R?ESUM?E D ans cette th?ese , nous nous int?eressons au probl?eme de la super-r ? esolution . Dans le premier chapitre , nous pr?esentons une ? etude comparative des m?ethodes propos?ees dans la litt?erature suivie d'une classification de ces m?ethodes du point de vue du crit?ere de Rayleigh et de celui d'Abbe sur la notion de r?esolution . Nous concluons le chapitre par une ? etude des diff?erentes classes de m?ethodes dans le cadre de la notion d'invariant optique . Dans le deuxi?eme chapitre le probl?eme de la super-r ? esolution multi-canal est ? etudi?e dans le cadre de la th?eorie des sous-espaces hilbertiens . Cette ? etude est fond?ee sur notre g?en?eralisation du th?eor?eme d' ? echantillonnage de Papoulis , qui nous permet d' ? etablir l'existence de la solution exacte dans le cas id?eal et de la meilleure approximation dans un sous-espace de Paley-Wiener quand le probl?eme est sous-d ? etermin?e ou mal pos?e . L'algorithme de reconstruction qui en r?esulte consiste ? a optimiser une fonction de co ^ ut par une projection dans un sous-espace r?egulier , ie . optimisation sous contraintes . Dans le troisi?eme chapitre , nous consid?erons d'abord le probl?eme de la mise en correspondance sous-pix ? elique des images . Cette ? etude est essentielle afin de pouvoir appliquer notre m?ethode aux donn?ees r?eelles . Ensuite nous abordons la question de faisabilit?e de la super-r ? esolution multi-canal qui d?epend de la stabilit?e de l' ? echantillonnage d?efinie par l'in ? egalit?e de Bessel . Enfin , nous ? etudions le probl?eme du bruit . Notre estimation est fond?ee sur le seuillage de la partie principale de la suite de Riemann d'une fonction contenant des irr?egularit?es dues au bruit . Nous avons d?emontr?e que les singularit?es isol?ees peuvent ^ etre s?epar?ees par un simple filtrage . Dans les deux ann?exes qui suivent , nous consid?erons la relation entre notre th?eor?eme d' ? echantillonnage et celui de Papoulis et nous ? etablissons aussi une g?en?eralisation de la m?ethode du simplex non-lin ? eaire . Mots clefs : vision par ordinateur , vision pr?e - attentive , super-r ? esolution , ? echantillonnage multi-canal , espace de Hilbert , mise en correspondance sous-pix ? elique , estimation ponctuelle du bruit , extraction de singularit?e , calcul fractionnaire ABSTRACT I n this thesis , we investigate the problem of super-resolution . In the first chapter , a comparative study of the existing methods is presented , which leads to the classification of these methods according to Rayleigh 's and Abbe 's criteria for resolvability . The chapter is concluded by unifying the different classes of methods using the notion of optical invariant . In the second chapter the problem of multichannel super-resolution is investigated in a Hilbert space setting . The backbone of our discussion is our generalization of Papoulis ' sampling theorem which allows us to establish the existence of both exact and best approximate solutions in Paley-Wiener subspaces . The resulting algorithm is based on optimizing a cost function by projecting onto regular Sobolev subspaces , ie . a constrained optimization . In the third chapter , we first investigate the problem of registering an image sequence at sub-pixel accuracy . This is essential for the application of our super-resolution method to real data . We then consider the problem of estimating the feasibility margins of multichannel super-resolution , in terms of the sampling stability defined by Bessel 's inequality . Finally the problem of estimating the noise process is formulated as that of thresholding out the principle part of the Riemann series expansion of a function , which contains isolated singularities due to added noise . We have demonstrated that this can be achieved by filtering in the Fourier domain . Two main appendices are also included . One which demonstrates the relation between our sampling theorem and that of Papoulis and the second which establishes a generalization of the simplex search method .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

computer vision
early vision
fractional calculus
hilbert space analysis
multichannel sampling
pointwise noise estimation
singularity extraction
sub-pixel image registration
super-resolution


Dynamic

analysis

of

some

Relational

Data

Bases

parameters

I

Projections

Dani?ele

GARDY

Guy

LOUCHARDy

January

Abstract

We

present

a

dynamic

study

of

a

data

structure

related

to

relational

databases

We

show

that

some

parameters

of

relational

databases

-LRB-

sizes

of

projections

-RRB-

related

to

an

occupancy

problem

in

urn

models

behave

asymptotically

as

gaussian

stochastic

processes

under

a

sequence

of

updates

and

queries

As

a

consequence

we

analyze

the

distribution

of

the

maximum

size

of

the

projection

Introduction

We

consider

dynamic

objects

obtained

by

updating

and

querying

a

data

structure

on

which

we

want

to

study

a

parameter

most

often

defining

some

size

This

parameter

defines

a

random

variable

and

we

study

its

behaviour

when

the

initial

object

is

submitted

to

a

sequence

of

insertions

deletions

and

queries

characterizing

it

-LRB-

when

possible

-RRB-

as

a

gaussian

stochastic

process

The

dynamic

objects

are

here

relations

in

a

relational

database

and

the

parameter

we

want

to

study

is

the

size

of

their

projection

on

a

set

of

attributes

We

gave

in

a

former

paper

conditions

which

ensure

that

in

the

static

case

-LRB-

i.e.

at

a

given

time

-RRB-

the

size

of

the

projection

of

a

relation

follows

a

normal

limiting

distribution

Our

goal

here

is

to

study

the

variation

of

the

size

of

the

projection

under

a

sequence

of

queries

and

updates

We

shall

show

that

it

is

a

gaussian

process

and

analyze

its

maximum

Laboratoire

PRISM

Universit?e

de

Versailles

Saint-Quentin

Versailles

-LRB-

France

-RRB-

This

research

was

partly

supported

by

ESPRIT

III-Basic

Research

Action

ALCOM

II

-LRB-

no.

-RRB-

by

the

CNRS

PRC

Math?ematique

Informatique

and

by

a

cooperation

between

the

CNRS

and

the

FNRS

yD?epartement

d'Informatique

Universit?e

Libre

de

Bruxelles

Bruxelles

-LRB-

Belgique

-RRB-

This

research

was

partially

supported

by

a

cooperation

between

the

FNRS

and

the

CNRS

R

X

Y

ssX

-LRB-

R

-RRB-

X

Figure

Projection

of

the

relation

R

-LSB-

X

Y

-RSB-

on

the

attribute

X

The

paper

is

organized

as

follows

Section

presents

the

database

parameters

that

we

shall

study

and

gives

a

modelization

in

terms

of

urn

models

then

briefly

recalls

the

sequences

of

operations

which

may

be

considered

Section

gives

our

main

results

-LRB-

characterization

of

the

parameter

we

study

as

a

gaussian

process

and

distribution

of

its

maximum

-RRB-

and

presents

an

overview

of

our

method

with

a

sketch

of

the

proof

Section

introduces

our

notations

then

Section

presents

the

basic

processes

-LRB-

number

of

tuples

in

a

relation

-RRB-

corresponding

to

different

update

models

and

to

several

constraints

on

the

initial

objects

-LRB-

relations

-RRB-

Sections

to

are

devoted

to

the

detailed

proofs

Databases

and

urn

models

Projections

and

the

occupancy

problem

in

urn

models

We

briefly

recall

here

some

definitions

relative

to

relational

databases

and

to

the

modelization

of

relations

we

refer

the

reader

to

-LSB-

-RSB-

for

a

detailed

presentation

The

basic

objects

we

consider

are

relations

which

are

sets

of

-LRB-

distinct

-RRB-

tuples

They

can

be

seen

as

tables

a

row

represents

a

tuple

and

the

number

of

lines

is

the

number

of

elements

of

the

relation

-LRB-

its

size

-RRB-

the

columns

are

called

the

attributes

The

projection

of

a

relation

on

a

subset

of

the

set

of

attributes

is

a

new

relation

obtained

by

suppressing

the

corresponding

columns

then

all

the

duplicate

rows

in

the

resulting

table

We

keep

only

one

instance

of

each

tuple

We

give

in

Figure

an

instance

of

a

relation

R

-LSB-

X

Y

-RSB-

and

of

its

projection

-LRB-

noted

ssX

-LRB-

R

-RRB-

-RRB-

on

the

attribute

X.

For

ease

of

presentation

and

without

loss

of

generality

we

shall

restrict

ourselves

to

the

case

of

a

relation

R

with

two

attributes

X

and

Y

and

of

its

projection

on

X

We

shall

use

the

terms

initial

relation

for

the

relation

R

and

derived

relation

for

its

projection

Let

d

be

the

number

of

distinct

possible

values

for

the

attribute

X

we

assume

that

although

it

may

become

large

d

is

finite

The

projection

of

the

relation

R

can

be

modelized

with

urns

and

balls

according

to

a

well-known

occupancy

model

as

follows

We

consider

a

sequence

of

d

urns

each

urn

being

labelled

with

a

distinct

value

of

the

attribute

X.

To

each

tuple

of

the

relation

R

we

associate

a

ball

labelled

by

the

value

of

the

tuple

on

the

column

X

this

ball

falls

into

the

corresponding

urn

An

equivalent

way

of

seeing

this

phenomenon

is

to

consider

instead

that

we

have

a

finite

supply

of

balls

and

that

we

allocate

them

at

random

among

the

d

urns

each

trial

being

independent

of

the

others

Each

ball

then

receives

the

label

of

the

urn

it

falls

into

After

coupling

all

the

tuples

of

the

initial

relation

R

with

urns

some

urns

are

empty

and

some

contain

at

least

one

ball

The

number

of

urns

with

at

least

one

ball

is

exactly

the

number

of

tuples

in

the

projection

of

the

relation

R.

If

instead

of

the

number

of

urns

with

at

least

one

ball

we

consider

the

number

of

empty

urns

and

if

we

assume

that

each

urn

can

receive

an

unbounded

number

of

balls

then

we

have

the

classical

occupancy

problem

presented

for

example

in

-LSB-

-RSB-

Assuming

that

the

urn

size

is

infinite

corresponds

in

terms

of

relational

databases

to

a

relation

with

a

key

on

the

attribute

suppressed

in

the

projection

As

we

shall

also

want

to

study

relations

without

keys

we

shall

have

to

extend

the

models

to

the

case

where

the

urns

have

a

finite

capacity

-LRB-

there

are

ffi

places

for

balls

-RRB-

More

generally

if

we

want

to

allow

for

constraints

on

the

database

relations

we

have

to

introduce

related

constraints

on

the

way

balls

can

be

allocated

into

urns

-LRB-

see

-LSB-

-RSB-

-RRB-

We

shall

make

the

following

assumptions

in

the

present

work

which

cover

a

reasonable

number

of

situations

while

keeping

the

computations

manageable

We

shall

assume

that

the

possible

values

for

the

projection

attribute

X

are

uniformly

distributed

and

that

when

the

attribute

Y

suppressed

by

the

projection

is

not

a

key

of

the

relation

the

possible

values

of

Y

are

also

uniformly

distributed

-LRB-

when

the

attribute

Y

is

key

of

the

relation

we

have

shown

that

the

probability

distribution

of

the

values

has

no

influence

on

the

size

of

the

result

-LSB-

-RSB-

-RRB-

In

the

rest

of

the

paper

we

shall

use

indifferently

the

terms

relation

size

and

number

of

balls

or

number

of

tuples

and

the

terms

projection

size

and

number

of

non-empty

urns

Dynamic

models

The

urn

model

we

have

just

defined

describes

well

a

relation

at

a

given

time

but

it

does

not

take

into

account

its

evolution

during

a

sequence

of

updates

and

queries

We

now

introduce

a

modelization

that

allows

us

to

consider

the

evolution

of

a

relation

subjected

to

a

sequence

of

updates

-LRB-

insertions

and

deletions

-RRB-

and

searches

-LRB-

queries

-RRB-

We

shall

denote

by

pI

pD

and

pQ

the

probability

of

making

an

insertion

or

a

deletion

or

a

query

If

these

probabilities

vary

according

to

the

time

t

we

shall

use

the

notations

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

and

pQ

-LRB-

t

-RRB-

We

can

choose

non-equal

probabilities

for

insertion

and

deletion

as

long

as

the

probability

of

an

insertion

is

at

least

equal

to

the

probability

of

a

deletion

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

Otherwise

the

relation

is

either

empty

or

has

very

few

elements

and

this

is

of

little

interest

both

in

terms

of

database

and

for

the

underlying

probability

model

We

must

now

make

precise

the

individual

probabilities

of

insertion

at

a

given

place

and

of

deletion

of

a

given

ball

If

we

choose

to

do

a

deletion

the

conditional

probability

of

deleting

a

given

ball

is

number

of

balls

at

this

time

both

for

the

infinite

urn

and

for

the

bounded

urn

models

If

we

choose

to

do

an

insertion

we

must

give

the

conditional

probability

of

inserting

a

ball

into

an

urn

and

this

is

the

place

where

the

infinite

and

finite

models

differ

In

the

infinite

urn

model

each

urn

has

the

same

probability

of

getting

the

new

ball

If

the

urns

are

bounded

we

can

view

each

urn

as

a

collection

of

ffi

distinguishable

cells

and

each

empty

cell

whatever

the

urn

it

belongs

to

has

the

same

conditional

probability

of

receiving

the

ball

knowing

that

we

have

chosen

to

do

an

insertion

To

fully

specify

the

dynamic

evolution

of

the

relation

we

also

specify

its

status

at

the

beginning

and

at

the

end

of

the

sequence

of

updates

and

queries

We

shall

assume

that

the

relation

is

empty

at

the

beginning

If

we

impose

a

condition

on

the

relation

at

the

end

-LRB-

this

is

not

mandatory

see

Section

for

such

examples

-RRB-

either

the

relation

is

empty

or

its

size

is

proportional

to

the

time

elapsed

Main

ideas

and

results

Our

goal

here

is

to

study

the

variation

of

the

size

of

the

projection

under

a

sequence

of

queries

and

updates

We

shall

do

this

during

a

time

intervall

of

length

for

large

n

-LRB-

n

is

a

scaling

factor

and

a

time

j

can

be

written

after

normalizing

as

j

nt

-RRB-

We

shall

study

two

related

stochastic

processes

describing

respectively

the

number

of

balls

denoted

by

P

and

the

size

of

the

projection

-LRB-

number

of

non-empty

urns

-RRB-

denoted

by

Q

we

shall

show

that

each

of

these

processes

has

a

deterministic

component

of

order

n

and

a

random

component

of

order

pn

Our

main

result

is

thus

the

following

theorem

where

the

functions

G

and

NR

can

be

given

explicitely

for

the

different

models

Theorem

The

size

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

of

the

projection

at

the

time

nt

is

asymptotically

a

nonMarkovian

Gaussian

process

such

that

E

-LSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-RSB-

nG

-LRB-

t

-RRB-

COV

-LRB-

S

-LRB-

-LSB-

-RSB-

-RRB-

S

-LRB-

-LSB-

-RSB-

-RRB-

-RRB-

n?R

-LRB-

-RRB-

V

AR

-LSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-RSB-

n

-LRB-

t

-RRB-

The

relative

error

in

the

density

is

O

-LRB-

pn

-RRB-

The

first

step

of

the

proof

is

to

study

the

process

P

describing

the

number

of

tuples

in

the

initial

relation

To

fully

describe

P

we

have

to

know

the

probabilities

for

insertion

deletion

and

query

and

to

give

the

initial

and

final

sizes

of

the

relation

In

the

cases

we

are

interested

in

we

can

show

that

P

is

a

gaussian

process

with

a

deterministic

part

on

which

is

superimposed

a

random

part

P

The

process

follows

a

deterministic

curve

-LRB-

t

-RRB-

the

function

is

related

to

the

exact

relation

-LRB-

or

urn

-RRB-

model

and

can

be

computed

explicitely

The

process

is

a

markovian

gaussian

process

of

order

pn

The

computation

of

and

and

of

several

related

parameters

such

as

is

done

in

Section

The

process

P

number

of

tuples

determines

another

process

Q

size

of

the

projection

Before

considering

Q

we

shall

study

another

process

defined

as

the

size

of

the

projection

of

a

relation

R

when

the

size

of

R

is

given

by

the

process

-LRB-

which

is

a

first-order

approximation

of

P

-RRB-

To

this

effect

we

define

two

random

variables

say

and

which

are

simply

the

size

of

the

projection

at

different

times

and

We

know

from

previous

work

-LSB-

-RSB-

the

conditional

expectation

and

variance

of

the

projection

size

for

a

given

relation

size

and

that

the

conditional

distribution

of

the

projection

size

follows

asymptotically

a

normal

distribution

The

covariance

COV

-LRB-

-RRB-

will

allow

us

to

characterize

as

a

process

composed

of

a

deterministic

part

nG

-LRB-

t

-RRB-

and

a

random

part

pn

V

-LRB-

t

-RRB-

The

computation

of

COV

-LRB-

-RRB-

starts

with

Lemma

of

Section

and

depends

on

the

stochastic

behaviour

of

the

number

of

balls

in

any

one

urn

This

behaviour

can

itself

be

expressed

both

for

the

bounded

and

for

the

infinite

urn

models

in

terms

of

the

probabilities

pI

pD

and

pQ

and

of

the

function

related

to

the

expectation

of

the

number

of

balls

-LRB-

Section

-RRB-

For

any

of

the

processes

of

Section

we

could

then

specialize

these

results

to

get

the

covariance

of

and

see

Sections

and

for

examples

of

such

computations

We

shall

rather

show

that

there

exists

a

common

form

giving

the

covariance

in

terms

of

pI

pD

and

pQ

this

is

Proposition

of

Section

We

then

consider

the

process

P

obtained

by

superimposing

on

We

can

again

define

two

random

variables

size

of

the

projection

at

the

times

and

let

us

call

them

and

As

we

have

done

for

and

we

have

to

compute

their

covariance

But

the

Si

are

obtained

from

the

Yi

by

introducing

a

further

degree

of

randomness

and

it

is

possible

to

write

their

covariance

as

Cov

-LRB-

-RRB-

Cov

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

for

a

suitable

function

-LRB-

t

-RRB-

-LRB-

-RRB-

being

the

covariance

of

the

process

taken

at

different

times

and

The

covariance

of

and

thus

characterizes

the

static

part

and

the

term

added

to

it

to

get

the

covariance

of

and

comes

from

the

fact

that

the

number

of

tuples

P

is

itself

a

gaussian

process

The

introduction

of

and

the

computation

of

COV

-LRB-

-RRB-

are

found

in

Section

Once

we

have

the

covariance

of

the

sizes

of

the

projection

at

times

and

i.e.

of

and

the

next

part

is

to

show

that

the

final

process

size

of

projection

which

we

denote

by

Q

is

still

asymptotically

a

gaussian

process

More

precisely

we

shall

show

in

Section

that

Q

has

a

part

coming

from

on

which

is

added

a

random

part

coming

from

and

from

Q

When

we

have

an

-LRB-

asymptotic

-RRB-

expression

for

the

covariance

of

and

and

are

sure

that

the

final

process

Q

is

gaussian

we

can

obtain

whatever

information

we

are

interested

in

We

shall

study

here

the

process

giving

the

maximum

size

of

the

projection

We

obtain

the

following

result

which

is

proved

in

Section

Theorem

Let

t

such

that

-LRB-

t

-RRB-

Assume

that

t

-RSB-

-LSB-

and

set

G

G

-LRB-

t

-RRB-

The

maximum

size

of

the

projection

M

max

-LSB-

-RSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

occurs

at

a

time

t

and

is

such

that

M

n

G

p

nm

O

-LRB-

-RRB-

where

m

and

t

are

random

variables

that

can

be

precisely

characterized

Notations

ffl

Let

n

be

some

scaling

parameter

-LRB-

n

later

on

-RRB-

ffl

We

have

d

urns

numbered

i

j

the

urn

numbered

i

is

Ui

The

parameters

n

and

d

are

related

by

d

-LRB-

n

-RRB-

ffn

say

ffl

When

the

capacity

of

an

urn

is

finite

it

is

denoted

by

ffi

-LRB-

constant

-RRB-

Let

dffi

n

with

ffffi

is

the

maximal

number

of

balls

that

we

can

allocate

to

the

urns

ffl

Let

be

the

random

variable

number

of

balls

in

the

urn

Ui

at

time

and

E

-LSB-

-RSB-

similary

for

at

time

For

any

measurable

function

and

for

a

random

variable

we

can

define

a

new

random

variable

-LRB-<=-RRB-

Let

-LSB-

-RSB-

-LSB-

-LRB-

-RRB-

-RSB-

We

use

in

this

paper

the

particular

case

-LRB-<=-RRB-

I

-LRB-

-RRB-

ffl

Ei

j

is

the

expectation

E

-LSB-

-RSB-

and

similarly

for

Ei

j

-LSB-

-RSB-

E

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ffl

We

denote

by

and

the

number

of

tuples

of

the

initial

relation

i.e.

the

total

number

of

balls

at

the

times

and

ffl

I

D

Q

denote

respectively

an

insertion

a

deletion

or

a

query

Their

probabilities

at

the

time

t

are

respectively

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

and

pQ

-LRB-

t

-RRB-

-LRB-

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

pQ

-LRB-

t

-RRB-

-RRB-

ffl

We

note

by

-RRB-

the

weak

convergence

of

random

functions

in

the

space

of

all

rightcontinuous

functions

having

left

limits

and

endowed

with

the

Skorohod

metric

-LRB-

see

Billingsley

-LSB-

-RSB-

-RRB-

All

convergences

with

be

defined

for

n

The

process

P

related

to

the

total

number

of

balls

Let

W

be

the

number

of

balls

at

some

time

We

might

choose

the

current

number

of

steps

-LRB-

number

of

queries

or

updates

-RRB-

as

a

measure

for

the

time

which

would

then

belong

to

the

intervall

-LSB-

-RSB-

However

we

shall

study

the

asymptotic

behaviour

of

W

when

the

time

goes

to

infinity

and

it

is

interesting

to

change

the

time

scale

by

choosing

a

time

nt

for

t

-LSB-

-RSB-

and

to

normalize

the

random

variable

W

For

all

the

models

presented

below

the

number

of

tuples

W

has

an

expectation

and

a

variance

of

order

n

and

we

can

show

that

for

a

suitable

function

related

to

the

type

of

process

and

assuming

that

we

start

from

an

empty

structure

at

time

W

-LRB-

-LSB-

nt

-RSB-

-RRB-

-LRB-

t

-RRB-

pn

-RRB-

X

-LRB-

t

-RRB-

t

where

the

process

X

-LRB-

t

-RRB-

is

a

Markovian

Gaussian

process

whose

covariance

is

denoted

-LRB-

s

t

-RRB-

s

t

As

a

consequence

we

have

that

for

any

and

E

-LSB-

ei

-LRB-

-RRB-

-RSB-

exp

n

ae

i

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

h

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

ioe

We

now

turn

to

the

presentation

of

the

models

we

shall

study

The

processes

can

be

divided

in

two

families

i

-RRB-

the

weighted

structure

in

the

sense

of

Flajolet

et

al

-LSB-

-RSB-

Louchard

-LSB-

-RSB-

with

a

possibility

function

given

by

pos

-LRB-

D

-RRB-

k

for

a

k-size

structure

-LRB-

there

are

k

ways

of

deleting

an

element

in

a

structure

composed

from

k

elements

-RRB-

ii

-RRB-

the

classical

unweighted

structure

In

the

weighted

structure

family

we

have

for

instance

ffl

I

D

we

assume

that

we

return

to

an

empty

structure

at

time

Then

-LSB-

-RSB-

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-LRB-

s

t

-RRB-

-LRB-

t

-RRB-

pI

-LRB-

t

-RRB-

-LSB-

f

-LRB-

t

-RRB-

-RSB-

t

pD

-LRB-

t

-RRB-

-LSB-

f

-LRB-

t

-RRB-

-RSB-

t

ffl

I

D

we

assume

that

we

return

to

a

structure

with

size

an

at

time

The

techniques

we

used

in

-LSB-

-RSB-

lead

here

to

-LRB-

t

-RRB-

t

-LRB-

a

-RRB-

t

-LRB-

s

t

-RRB-

a

-LRB-

t

-RRB-

-LRB-

t

at

-RRB-

s

t

Note

that

possess

a

maximum

for

t

-RSB-

-LSB-

iff

a

For

a

is

maximized

at

t

Again

pI

-LRB-

t

-RRB-

-LSB-

f

-LRB-

t

-RRB-

-RSB-

pD

-LRB-

t

-RRB-

-LSB-

f

-LRB-

t

-RRB-

-RSB-

ffl

We

should

be

tempted

to

extend

the

weighted

model

to

the

case

with

Q.

But

when

P

-LRB-

insertion

-RRB-

P

-LRB-

deletion

-RRB-

and

P

-LRB-

query

-RRB-

we

see

by

-LSB-

Lemma

-RSB-

that

along

a

structure

with

mean

-LRB-

t

-RRB-

we

have

the

following

probabilities

after

weighting

pI

-LRB-

t

-RRB-

h

-LRB-

t

-RRB-

f

-LRB-

t

-RRB-

O

-LRB-

n

-RRB-

pD

-LRB-

t

-RRB-

h

-LRB-

t

-RRB-

f

-LRB-

t

-RRB-

O

-LRB-

n

-RRB-

pQ

-LRB-

t

-RRB-

h

f

-LRB-

t

-RRB-

O

-LRB-

n

-RRB-

But

now

the

logarithm

of

the

asymptotic

total

measure

along

-LRB-

t

-RRB-

contains

a

dominant

term

-LRB-

see

-LSB-

Lemma

-RSB-

-RRB-

Z

n

log

n

f

which

by

Euler-Lagrange

is

maximized

by

f

C.

With

constraints

on

the

structure

this

leads

to

f

For

instance

with

-LRB-

-RRB-

a

this

gives

a

trajectory

of

type

-LRB-

no

queries

at

all

-RRB-

Figure

Weighted

model

with

Q

which

is

a

completely

uninteresting

process

So

we

turn

to

the

unweighted

structure

family

ffl

I

-LRB-

pI

-RRB-

We

have

and

-LRB-

t

-RRB-

t.

ffl

I

D

Q

with

pI

pD

and

pQ

constant

-LRB-

pI

pQ

-RRB-

and

without

constraint

on

the

relation

size

at

the

time

The

mean

and

variance

corresponding

to

one

step

are

given

by

x

pI

pD

pI

pD

So

-LRB-

t

-RRB-

xt

-LRB-

s

t

-RRB-

s

t

This

is

a

classical

Brownian

Motion

-LRB-

BM

-RRB-

ffl

I

D

Q

with

arrival

at

a

relation

of

size

x

apn

at

the

time

The

mean

x

and

variance

corresponding

to

one

step

are

the

same

as

for

and

W

-LRB-

-LSB-

nt

-RSB-

-RRB-

n?xt

pn

-RRB-

oe

BB

-LRB-

t

-RRB-

at

with

BB

a

Brownian

Bridge

The

expectation

and

covariance

of

W

are

given

by

-LRB-

t

-RRB-

xt

at

-LRB-

s

t

-RRB-

s

-LRB-

t

-RRB-

s

t

ffl

I

D

Q

with

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

pQ

-LRB-

t

-RRB-

time

dependent

The

infinitesimal

mean

and

variance

are

given

by

x

-LRB-

s

-RRB-

pI

-LRB-

s

-RRB-

pD

-LRB-

s

-RRB-

-LRB-

s

-RRB-

pI

-LRB-

s

-RRB-

pD

-LRB-

s

-RRB-

-LRB-

s

-RRB-

so

-LRB-

t

-RRB-

Z

t

o

x

-LRB-

s

-RRB-

ds

-LRB-

t

-RRB-

Z

t

o

-LRB-

s

-RRB-

ds

This

is

a

time-dependent

BM

which

can

be

written

as

-LRB-

t

-RRB-

Z

t

-LRB-

s

-RRB-

-LRB-

s

-RRB-

where

and

are

standard

BM

ffl

with

the

finite

urn

case

assume

that

each

position

taken

at

random

among

the

possible

positions

changes

from

status

-LRB-

full

empty

empty

full

-RRB-

This

is

equivalent

to

the

Ehrenfest

urn

model

-LRB-

I

D

-RRB-

From

Karlin

and

Taylor

-LSB-

p.

-RSB-

we

know

that

if

then

W

-LRB-

-LSB-

Nt

-RSB-

-RRB-

N

p

N

-RRB-

OU

-LRB-

t

-RRB-

-LRB-

OU

-LRB-

-RRB-

if

W

-LRB-

-RRB-

N

-RRB-

where

OU

is

the

classical

Ornstein-Uhlenbeck

process

with

mean

and

covariance

-LSB-

e

-LRB-

t?s

-RRB-

e

-LRB-

t

s

-RRB-

-RSB-

s

t.

Note

that

this

covariance

rapidly

converges

to

its

stationary

form

-LRB-

t?s

-RRB-

as

s

t

t

s

O

-LRB-

-RRB-

Here

n

so

N

n

-LRB-

t

-RRB-

and

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

If

we

start

with

W

-LRB-

-RRB-

n

then

W

-LRB-

-LSB-

nt

-RSB-

-RRB-

n

pn

-RRB-

s

OU

with

covariance

-LRB-

s

t

-RRB-

h

e

-LRB-

t?s

-RRB-

fi

e

-LRB-

t

s

-RRB-

fii

If

we

start

with

W

-LRB-

-RRB-

then

it

is

easily

checked

that

W

-LRB-

-LSB-

nt

-RSB-

-RRB-

n

pn

p

ne

fi

-RRB-

s

OU

and

-LRB-

t

-RRB-

-LRB-

e

fi

-RRB-

Urn

models

-LRB-

A

and

B

-RRB-

preliminary

results

General

form

of

the

covariance

Lemma

Let

be

the

sizes

of

the

projection

at

times

Lemma

below

gives

a

general

expression

for

their

covariance

COV

-LRB-

-RRB-

in

terms

of

some

probabilities

that

can

be

defined

whatever

the

urn

model

We

recall

that

-LRB-

resp

-RRB-

is

a

random

variable

giving

the

number

of

balls

in

the

urn

Ui

at

the

time

-LRB-

resp

-RRB-

Define

-LRB-<=-RRB-

I

-LRB-

-RRB-

then

the

projections

sizes

and

at

the

times

and

can

be

written

as

Pdi

-LRB-

-RRB-

and

Pdj

-LRB-

-RRB-

Let

Z

P

r

-LSB-

-RSB-

E

-LSB-

I

-LRB-

-RRB-

-RSB-

-LRB-

-RRB-

and

similarly

for

Z

Z

-LRB-

resp

Z

-RRB-

is

the

probability

that

the

urn

Ui

is

empty

at

the

time

-LRB-

resp

-RRB-

Define

also

the

joint

probability

Z

i

j

P

r

-LSB-

-RSB-

-LRB-

-RRB-

The

term

Z

i

j

is

the

probability

that

the

urn

Ui

is

empty

at

the

time

and

that

the

urn

Uj

is

empty

at

the

time

Lemma

The

expectation

of

the

projection

size

at

the

time

and

the

covariance

of

the

projection

sizes

and

at

the

times

and

can

be

expressed

in

terms

of

the

probabilities

Z

and

Z

i

j

defined

by

-LRB-

-RRB-

and

-LRB-

-RRB-

E

-LRB-

-RRB-

d

-LRB-

Z

-RRB-

COV

-LRB-

-RRB-

d

-LRB-

Z

i

i

Z

i

j

-RRB-

i

j

with

C

i

j

Z

i

j

Z

Proof

As

the

urns

are

equiprobable

E

-LRB-

-RRB-

d

X

i

E

-LSB-

-LRB-

-RRB-

-RSB-

d

X

i

-LSB-

-RSB-

-LSB-

-RSB-

Similarly

E

-LRB-

-RRB-

-LSB-

-RSB-

The

probability

that

the

urn

Ui

is

not

empty

at

time

is

-LSB-

-RSB-

and

the

probability

that

the

urn

Uj

is

not

empty

at

time

is

-LSB-

-RSB-

-LSB-

-RSB-

Z

-LSB-

-RSB-

By

definition

COV

-LRB-

-RRB-

E

-LRB-

-RRB-

E

-LRB-

-RRB-

E

-LRB-

-RRB-

and

E

-LRB-

-RRB-

d

X

i

j

E

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

X

i

Ei

i

-LSB-

-RSB-

X

j

Ei

j

-LSB-

-RSB-

dEi

i

-LSB-

-RSB-

d

-LRB-

d

-RRB-

Ei

j

-LSB-

-RSB-

Now

Ei

j

-LSB-

-RSB-

is

the

joint

probability

that

the

urn

Ui

is

not

empty

at

time

and

that

the

urn

Uj

is

not

empty

at

time

By

an

argument

of

inclusion-exclusion

we

get

Ei

j

-LSB-

-RSB-

P

r

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

P

r

-LRB-

-LRB-

-RRB-

-RRB-

P

r

-LRB-

-LRB-

-RRB-

-RRB-

P

r

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

Z

Z

i

j

Hence

COV

-LRB-

-RRB-

dEi

i

-LSB-

-RSB-

d

-LRB-

d

-RRB-

Ei

j

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

d

-LRB-

Z

Z

i

i

Z

i

j

-RRB-

-LRB-

Z

i

j

Z

-RRB-

By

symmetry

Z

which

gives

the

desired

result

Let

us

now

turn

to

the

urn

models

The

stochastic

behaviour

of

a

specific

urn

depends

only

on

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

pQ

-LRB-

t

-RRB-

and

-LRB-

t

-RRB-

which

depend

in

turn

on

the

type

of

process

we

are

concerned

with

-LRB-

see

Section

-RRB-

We

shall

express

our

results

in

terms

of

the

parameters

pI

pD

pQ

and

first

for

infinite

urns

then

for

finite

urns

These

results

can

then

be

specified

for

a

given

process

i.e.

for

a

choice

of

pI

pD

pQ

and

Model

A

the

urns

are

of

unlimited

size

We

recall

that

the

number

of

urns

d

and

the

parameter

n

are

related

by

d

ffn

and

that

the

average

number

of

balls

at

the

time

nt

is

-LRB-

t

-RRB-

Following

Louchard

-LSB-

-RSB-

we

see

that

asymptotically

the

number

of

balls

in

one

urn

is

given

by

a

classical

birth

and

death

process

with

rates

-LRB-

t

-RRB-

pI

-LRB-

t

-RRB-

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

-LRB-

t

-RRB-

The

one

ball

survival

probability

between

and

is

given

by

-LRB-

-RRB-

exp

Z

-LRB-

s

-RRB-

ds

-LRB-

if

-RRB-

The

total

number

of

balls

inserted

in

one

urn

between

and

which

have

not

been

deleted

at

follows

a

Poisson

distribution

with

parameter

-LRB-

-RRB-

ff

with

-LRB-

-RRB-

Z

pI

-LRB-

u

-RRB-

-LRB-

u

-RRB-

du

-LRB-

if

-RRB-

Note

that

by

obvious

probabilistic

reasoning

we

have

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Model

B

the

urns

are

bounded

In

order

to

avoid

trivial

models

we

assume

that

the

average

number

of

tuples

does

not

exceed

the

maximal

capacity

of

all

the

urns

in

the

time

interval

we

consider

-LRB-

t

-RRB-

i.e.

-LRB-

t

-RRB-

We

first

prove

two

lemmae

giving

the

number

of

balls

in

an

urn

at

the

time

and

its

evolution

between

and

then

analyze

the

conditional

distribution

of

the

number

of

balls

in

an

urn

at

the

time

conditioned

by

the

number

of

balls

in

the

urn

at

the

time

Distribution

of

the

number

of

balls

in

an

urn

Lemma

At

the

time

the

number

of

balls

in

the

urn

Ui

is

asymptotically

Binomial

-LRB-

Bin

-RRB-

with

parameters

ffi

-LRB-

-RRB-

fi

Proof

The

probability

that

there

are

k

balls

in

the

urn

Ui

is

P

r

-LSB-

k

-RSB-

ffi

k

ffi

ffi

k

-LRB-

ffi

-RRB-

-LRB-

-RRB-

-LRB-

ffi

k

-RRB-

-LRB-

k

-RRB-

Now

for

fixed

k

and

ffi

-LRB-

k

-RRB-

-LRB-

ffi

-RRB-

ffi

-LRB-

-RRB-

-LRB-

ffi

k

-RRB-

-LRB-

-RRB-

ffi?k

Hence

P

r

-LSB-

k

-RSB-

ffi

k

-LRB-

-RRB-

ffi?k

ffi

and

as

n

and

-LRB-

-RRB-

ffi

k

-LRB-

-RRB-

ffi?k

ffi

ffi

k

-LSB-

-LRB-

-RRB-

-RSB-

ffi?k

nffi?k

ffi

nffi

ffi

k

-LRB-

-RRB-

ffi?k

-LRB-

-RRB-

k

which

shows

that

asymptotically

follows

a

binomial

distribution

Lemma

Given

that

we

start

will

balls

in

the

urn

Ui

at

the

time

the

number

of

balls

-LRB-

t

-RRB-

-LRB-

t

-RRB-

in

the

urn

Ui

is

described

asymptotically

by

a

birth

and

death

process

starting

from

with

birth

rate

-LRB-

t

-RRB-

-LSB-

ffi

-LRB-

t

-RRB-

-RSB-

-LRB-

t

-RRB-

where

-LRB-

t

-RRB-

pI

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-RSB-

is

the

birth

rate

in

a

cell

and

individual

death

rate

-LRB-

t

-RRB-

with

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

-LRB-

t

-RRB-

Proof

The

probability

of

insertion

in

the

urn

Ui

between

t

and

t

n

is

given

by

pI

-LRB-

t

-RRB-

ffi

-LRB-

t

-RRB-

n

-LRB-

t

-RRB-

-LRB-

ffi

-LRB-

t

-RRB-

-RRB-

pI

-LRB-

t

-RRB-

-LSB-

-LRB-

t

-RRB-

-RSB-

n

Similarly

the

probability

that

one

tuple

is

deleted

in

the

urn

Ui

is

given

by

pD

-LRB-

t

-RRB-

-LRB-

t

-RRB-

n

-LRB-

t

-RRB-

-LRB-

t

-RRB-

n

pD

-LRB-

t

-RRB-

-LRB-

t

-RRB-

Conditional

distribution

To

analyze

the

distribution

of

the

number

of

balls

in

an

urn

at

the

time

conditioned

by

the

number

of

balls

in

the

same

urn

at

the

time

it

is

convenient

to

introduce

the

function

-LRB-

k

-RRB-

P

r

-LSB-

-LRB-

-RRB-

kj

-LRB-

-RRB-

-RSB-

We

can

see

the

content

of

an

urn

with

-LRB-

t

-RRB-

balls

as

a

population

of

-LRB-

t

-RRB-

type

-LRB-

balls

-RRB-

individuals

and

ffi

-LRB-

t

-RRB-

type

-LRB-

empty

places

-RRB-

individuals

changing

type

with

rate

and

by

Lemma

At

the

time

-LRB-

-RRB-

and

ffi

-LRB-

-RRB-

ffi

Let

pi

j

-LRB-

-RRB-

P

r

-LSB-

individual

of

type

i

at

the

time

is

of

type

j

at

the

time

-RSB-

So

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

the

number

of

balls

in

the

urn

at

the

time

is

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

where

-LRB-

-RRB-

is

the

number

of

balls

existing

at

the

time

which

are

still

alive

at

the

time

and

-LRB-

-RRB-

is

the

number

of

empty

places

at

the

time

which

contain

a

ball

at

the

time

The

random

variables

and

are

independent

with

distributions

Bin

-LRB-

-LRB-

-RRB-

-RRB-

and

Bin

-LRB-

ffi

-LRB-

-RRB-

-RRB-

The

probability

-LRB-

-RRB-

that

a

cell

which

is

empty

at

the

time

is

full

at

the

time

satisfies

-LRB-

dt

-RRB-

-LRB-

-RRB-

Pr

-LSB-

survival

during

dt

-RSB-

-LRB-

-RRB-

Pr

-LSB-

birth

during

dt

-RSB-

Now

the

probability

that

there

is

a

birth

in

an

empty

cell

during

an

intervall

of

time

dt

is

and

the

probability

that

the

individual

in

an

occupied

cell

survives

during

a

time

dt

is

hence

satisfies

the

relation

-LRB-

dt

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

dt

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

dt

Define

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-LRB-

t

-RRB-

-LRB-

-RRB-

exp

-LSB-

Z

-LRB-

v

-RRB-

dv

-RSB-

With

-LRB-

-RRB-

and

taking

dt

we

readily

obtain

that

the

function

t

-LRB-

t

-RRB-

is

solution

of

the

differential

equation

Y

-LRB-

t

-RRB-

-LRB-

t

-RRB-

Y

-LRB-

t

-RRB-

-LRB-

t

-RRB-

with

Y

-LRB-

-RRB-

Solving

we

get

-LRB-

-RRB-

Z

-LRB-

u

-RRB-

-LRB-

u

-RRB-

du

The

probability

-LRB-

-RRB-

that

a

cell

which

is

full

at

the

time

is

again

full

at

the

time

satisfies

the

same

differential

equation

but

with

the

initial

condition

Y

-LRB-

-RRB-

hence

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Z

-LRB-

u

-RRB-

-LRB-

u

-RRB-

du

Now

the

probability

-LRB-

-RRB-

that

a

cell

which

is

full

at

the

time

becomes

empty

at

the

time

satisfies

the

differential

equation

Y

-LRB-

t

-RRB-

-LRB-

t

-RRB-

Y

-LRB-

t

-RRB-

-LRB-

t

-RRB-

with

Y

-LRB-

-RRB-

so

-LRB-

-RRB-

Z

-LRB-

u

-RRB-

-LRB-

u

-RRB-

du

The

probability

-LRB-

-RRB-

that

a

cell

empty

at

is

still

empty

at

satisfies

the

same

equation

as

but

with

the

initial

condition

Y

-LRB-

-RRB-

hence

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Z

-LRB-

u

-RRB-

-LRB-

u

-RRB-

du

-LRB-

-RRB-

Finally

we

obtain

the

probability

-LRB-

k

-RRB-

that

an

urn

is

empty

at

the

time

knowing

that

it

contains

k

balls

at

the

time

The

k

balls

become

empty

places

and

the

ffi

k

empty

places

stay

empty

hence

-LRB-

k

-RRB-

-LSB-

-LRB-

-RRB-

-RSB-

k

-LSB-

-LRB-

-RRB-

-RSB-

ffi?k

-LRB-

-RRB-

The

conditional

expectation

of

the

number

of

balls

at

the

time

knowing

the

number

of

balls

at

the

time

is

-LRB-

k

-RRB-

E

-LSB-

-LRB-

-RRB-

j

-LRB-

-RRB-

k

-RSB-

-LRB-

-RRB-

-LRB-

ffi

k

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

The

average

number

of

balls

in

an

urn

at

the

time

t

is

-LRB-

t

-RRB-

=d

the

number

of

balls

in

an

urn

at

the

times

and

i.e.

-LRB-

-RRB-

=d

and

-LRB-

-RRB-

=d

are

related

by

-LRB-

-RRB-

d

-LRB-

-RRB-

d

-LRB-

-RRB-

-LRB-

ffi

-LRB-

-RRB-

d

-RRB-

-LRB-

-RRB-

Rewriting

we

get

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Computation

of

COV

-LRB-

-RRB-

for

a

non-random

static

structure

In

the

next

subsections

we

give

some

examples

of

the

computation

of

COV

-LRB-

-RRB-

before

dealing

with

the

general

case

in

Section

We

use

the

notations

P

i

A

for

the

process

P

i

in

the

model

with

infinite

urns

and

P

i

B

for

the

process

P

i

in

the

model

with

bounded

urns

A

unbounded

urns

and

insertions

The

probability

that

the

urn

Ui

contains

at

least

one

ball

at

the

time

and

at

the

time

is

simply

the

probability

that

the

urn

is

not

empty

at

the

time

-LRB-

there

are

no

deletions

-RRB-

Z

i

i

Z

and

Lemma

gives

COV

d

-LRB-

Z

Z

i

j

-RRB-

-LRB-

Z

i

j

Z

-RRB-

The

next

step

is

to

compute

the

probabilities

Z

and

Z

i

j

We

readily

derive

Z

d

ff

Z

d

The

term

Z

i

j

is

the

probability

that

the

urn

Ui

is

empty

at

the

time

and

that

the

urn

Uj

is

empty

at

the

time

As

there

are

no

deletions

this

is

also

the

probability

that

the

urns

Ui

and

Uj

are

empty

at

the

time

-LRB-

after

throwing

balls

-RRB-

and

that

the

urn

Uj

is

still

empty

at

the

time

after

throwing

new

balls

Z

i

j

d

d

d

d

Hence

COV

d

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

d

-RRB-

-LRB-

d

-RRB-

A

direct

asymptotic

analysis

leads

to

E

-LRB-

-RRB-

n

ffn

-LRB-

ff

-RRB-

COV

n

n

-LSB-

ff

ffe

-LRB-

-RRB-

ff

-LRB-

-RRB-

ff

-RSB-

-LRB-

-RRB-

However

for

further

application

it

is

more

convenient

to

rewrite

the

covariance

in

a

way

that

leads

to

computation

of

a

partial

derivative

We

use

the

simple

identity

=d

-LRB-

=d

-RRB-

-LRB-

-LRB-

d

-RRB-

-RRB-

to

get

COV

d

d

d

d

-LRB-

-RRB-

d

d

d

and

we

directly

analyze

the

second

bracket

term

of

-LRB-

-RRB-

Indeed

this

gives

an

equivalent

in

terms

of

a

partial

derivative

for

d

ffn

-LSB-

d

d

-RSB-

@d

-LRB-

d

-RRB-

ff

n

n

ff

which

immediately

leads

to

-LRB-

-RRB-

B

bounded

urns

and

insertions

In

this

case

the

number

of

balls

at

the

time

is

The

approximate

value

of

the

probability

Z

that

the

urn

Ui

is

empty

at

the

time

is

given

by

Lemma

with

-LRB-

t

-RRB-

t

Z

ffi

The

probability

that

the

urn

Ui

is

empty

at

the

time

and

at

the

time

is

again

simply

the

probability

that

it

is

empty

at

the

time

Z

i

i

ffi

The

joint

probability

that

the

urn

Ui

is

empty

at

the

time

and

that

the

urn

is

empty

at

the

time

is

Z

i

j

ffi

ffi

ffi

with

ffi

n

and

So

asymptotically

Z

i

j

ffi

ffi

By

C

i

j

Z

i

j

Z

we

obtain

C

i

j

ffi

ffi

ffi

the

bracket

leads

to

-LSB-

-RSB-

ffi

n

ffi

-LRB-

fi

-RRB-

ffi

and

C

i

j

ffi

-LRB-

fi

-RRB-

ffi

and

finally

with

Lemma

E

-LRB-

-RRB-

ffn

COV

n

ffi

ffi

ffi

ffi

Non-random

static

structure

We

now

extend

the

computations

of

Sections

and

to

the

general

cases

Let

us

start

with

a

non-random

-LRB-

NR

-RRB-

static

structure

where

the

process

total

number

of

tuples

follows

a

path

fixed

by

-LRB-

t

-RRB-

i.e.

the

path

is

the

closest

to

-LRB-

t

-RRB-

in

the

adequate

topology

-LRB-

Skorohod

for

instance

-RRB-

On

a

short

interval

Dt

such

that

nDt

each

step

choosen

at

random

among

m

has

probability

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

or

pQ

-LRB-

t

-RRB-

of

giving

an

insertion

a

deletion

or

a

query

All

the

results

in

this

section

depend

only

on

the

function

-LRB-

t

-RRB-

specifying

the

number

of

balls

at

time

t

on

the

probabilities

pI

-LRB-

t

-RRB-

pD

-LRB-

t

-RRB-

and

pQ

-LRB-

t

-RRB-

and

on

the

auxiliary

functions

defined

in

Sections

and

We

shall

show

that

the

average

value

of

the

projection

size

at

time

E

-LRB-

-RRB-

and

its

covariance

at

distinct

times

and

COV

-LRB-

-RRB-

both

have

a

common

form

whatever

the

model

Proposition

For

each

urn

model

A

or

B

there

exist

two

functions

F

-LRB-

x

-RRB-

and

NR

-LRB-

-RRB-

such

that

if

we

consider

the

size

of

the

projection

of

a

relation

itself

of

size

-LRB-

t

-RRB-

the

asymptotic

values

of

its

expectation

at

time

E

-LRB-

-RRB-

and

of

its

covariance

at

distinct

times

and

COV

-LRB-

-RRB-

are

E

-LRB-

-RRB-

nF

-LRB-

-LRB-

-RRB-

-RRB-

COV

-LRB-

-RRB-

n?NR

-LRB-

-RRB-

We

shall

study

two

examples

before

proving

Proposition

in

Section

Unbounded

urns

and

return

to

an

empty

structure

A

We

use

the

notations

Z

-LRB-

A

-RRB-

and

so

on

to

emphazise

the

dependence

on

the

urn

model

For

the

model

we

obtain

E

-LRB-

-RRB-

ffn

Z

-LRB-

A

-RRB-

ffn

-LRB-

-RRB-

ff

nF

-LRB-

-LRB-

-RRB-

-RRB-

say

with

-LRB-

t

-RRB-

t

-LRB-

t

-RRB-

By

Lemma

COV

-LRB-

-RRB-

ffn

h

Z

i

i

-LRB-

A

-RRB-

Z

i

j

-LRB-

A

-RRB-

i

i

j

-LRB-

A

-RRB-

So

we

obtain

COV

n

n

-LSB-

ffe

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

ffe

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

-LRB-

-RRB-

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

-RSB-

n?NR

-LRB-

-RRB-

and

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Bounded

urn

and

queries

B

We

obtain

E

-LRB-

-RRB-

ffn

Z

-LRB-

B

-RRB-

ffn

-LRB-

-RRB-

A

nF

-LRB-

-LRB-

-RRB-

-RRB-

ffn

A

COV

-LRB-

-RRB-

n?NR

-LRB-

-RRB-

ffn

h

Z

i

i

-LRB-

B

-RRB-

Z

i

j

-LRB-

B

-RRB-

i

i

j

-LRB-

B

-RRB-

ffn

ffi

-LRB-

-RRB-

ffi

n

x

-LRB-

-RRB-

ffi

ffi

Proof

of

Proposition

From

Lemma

of

Section

we

know

that

E

-LRB-

-RRB-

d

-LRB-

Z

-RRB-

nff

-LRB-

Z

-RRB-

The

computations

given

in

the

Appendix

-LRB-

see

Section

-RRB-

give

ffl

For

the

model

with

unbounded

urns

Z

-LRB-

t

-RRB-

ff

and

E

-LRB-

-RRB-

nF

-LRB-

-LRB-

t

-RRB-

-RRB-

with

F

-LRB-

x

-RRB-

-LRB-

e?x

ff

-RRB-

ffl

For

the

model

with

bounded

urns

Z

-LRB-

-LRB-

t

-RRB-

fi

-RRB-

ffi

and

E

-LRB-

-RRB-

nF

-LRB-

-LRB-

t

-RRB-

-RRB-

with

F

-LRB-

x

-RRB-

-LRB-

-LRB-

x

fi

-RRB-

ffi

-RRB-

Lemma

also

gives

an

expression

of

the

covariance

which

we

shall

combine

with

the

computations

of

Section

COV

-LRB-

-RRB-

d

-LRB-

Z

i

i

Z

i

j

-RRB-

i

j

ffl

For

the

model

with

unbounded

urns

COV

-LRB-

-RRB-

n

-LRB-

ffe

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

ff

ffe

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

ff

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

ff

-RRB-

hence

COV

-LRB-

-RRB-

n?NR

-LRB-

-RRB-

for

a

function

NR

-LRB-

-RRB-

which

is

expressed

in

terms

of

the

functions

-LRB-

t

-RRB-

and

-LRB-

-RRB-

NR

-LRB-

-RRB-

-LRB-

-RRB-

ff

-LRB-

-RRB-

ff

-LRB-

-RRB-

ff

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

ff

-LRB-

-RRB-

ffl

For

the

model

with

bounded

urns

and

using

the

functions

defined

in

Section

COV

-LRB-

-RRB-

n

-LRB-

-LRB-

-LRB-

-RRB-

-RRB-

ffi

-LSB-

-LRB-

-RRB-

ffi

-LRB-

-LRB-

-RRB-

-RRB-

ffi

-RSB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-RRB-

ffi

-LRB-

-LRB-

-RRB-

-RRB-

ffi

-RRB-

hence

COV

-LRB-

-RRB-

n?NR

-LRB-

-RRB-

for

a

suitable

function

NR

-LRB-

-RRB-

given

below

-LRB-

we

recall

that

and

are

functions

of

and

-RRB-

NR

-LRB-

-RRB-

-LRB-

-RRB-

ffi

-LRB-

-RRB-

ffi

-LRB-

ffi

-RRB-

-LRB-

-RRB-

Random

Structure

We

can

generalize

the

techniques

we

used

in

-LSB-

-RSB-

We

recall

that

and

denote

the

size

of

the

projection

of

a

relation

R

at

the

times

and

when

the

number

of

tuples

of

R

is

given

by

the

process

and

that

and

denote

the

same

quantities

when

the

number

of

tuples

of

R

is

given

by

the

process

P

We

first

compute

the

variation

of

COV

-LRB-

-RRB-

introduced

by

assuming

that

the

numbers

of

tuples

are

no

longer

fixed

but

gaussian

random

variables

this

gives

a

term

that

we

call

n?C

-LRB-

-RRB-

Then

we

compute

the

actual

covariance

of

and

and

we

show

that

it

is

of

the

type

n?R

-LRB-

-RRB-

we

also

prove

that

the

size

of

the

projection

is

then

a

gaussian

process

We

state

below

our

result

before

proving

it

in

the

rest

of

this

section

In

this

theorem

and

are

the

expectation

and

covariance

of

the

process

associated

with

the

initial

relation

and

-LRB-

t

-RRB-

F

-LRB-

-LRB-

t

-RRB-

-RRB-

Theorem

In

the

projection

model

the

size

of

the

projection

S

-LSB-

nt

-RSB-

is

a

non-Markovian

Gaussian

process

with

E

-LSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-RSB-

nG

-LRB-

t

-RRB-

with

G

-LRB-

t

-RRB-

F

-LRB-

-LRB-

t

-RRB-

-RRB-

COV

-LRB-

S

-LRB-

-LSB-

-RSB-

-RRB-

S

-LRB-

-LSB-

-RSB-

-RRB-

-RRB-

n?R

-LRB-

-RRB-

with

R

-LRB-

-RRB-

NR

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

V

AR

-LSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-RSB-

n

-LRB-

t

-RRB-

with

-LRB-

t

-RRB-

R

-LRB-

t

t

-RRB-

NR

-LRB-

t

t

-RRB-

-LRB-

t

-RRB-

-LRB-

t

t

-RRB-

The

relative

error

in

the

density

is

O

-LRB-

pn

-RRB-

The

perturbation

on

the

expectation

and

covariance

of

and

In

this

part

we

take

into

account

the

random

part

of

the

number

of

balls

W

-LRB-

-LSB-

nt

-RSB-

-RRB-

As

the

process

P

is

obtained

by

adding

a

process

of

order

pn

to

the

process

itself

of

order

n

we

have

n

-LRB-

-RRB-

pn

O

-LRB-

-RRB-

-LRB-

-RRB-

balls

at

and

n

-LRB-

-RRB-

pn

O

-LRB-

-RRB-

-LRB-

-RRB-

balls

at

where

and

are

Gaussian

random

variables

with

mean

and

covariance

-LRB-

s

t

-RRB-

for

any

and

E

-LSB-

ei

-LRB-

-RRB-

-RSB-

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

-LRB-

-RRB-

Set

-LRB-

t

-RRB-

F

-LRB-

-LRB-

t

-RRB-

-RRB-

-LRB-

-RRB-

From

Proposition

of

Section

we

have

that

E

-LSB-

-RSB-

nF

-LRB-

n

-RRB-

and

with

-LRB-

-RRB-

this

gives

E

-LSB-

-RSB-

n

F

-LRB-

-LRB-

-RRB-

-RRB-

pn

-LRB-

-RRB-

A

similar

relation

holds

for

E

-LSB-

-RSB-

Now

injecting

the

values

of

and

given

by

the

formulae

-LRB-

-RRB-

and

-LRB-

-RRB-

into

the

covariance

COV

-LRB-

-RRB-

n?NR

-LRB-

-RRB-

we

get

a

new

value

n?C

-LRB-

-RRB-

with

C

-LRB-

-RRB-

NR

-LRB-

-RRB-

-LRB-

-RRB-

pn

-LRB-

-RRB-

pn

O

n

for

some

and

and

for

NR

computed

in

Section

according

to

the

type

of

urn

The

covariance

of

and

We

know

from

previous

work

-LSB-

-RSB-

that

for

a

known

size

of

the

initial

relation

R

at

the

times

and

the

projection

sizes

and

are

asymptotically

gaussian

Then

for

any

and

E

h

ei

-LRB-

-RRB-

i

E

h

ei

-LRB-

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

i

Plugging

into

this

equation

the

modified

values

for

E

-LSB-

-RSB-

and

E

-LSB-

-RSB-

and

substituting

n?C

-LRB-

-RRB-

for

COV

-LRB-

-RRB-

-LRB-

and

similarly

for

-LRB-

-RRB-

and

-LRB-

-RRB-

-RRB-

we

obtain

an

expression

for

E

h

ei

-LRB-

-RRB-

i

E

h

ei

-LRB-

-RRB-

i

eA

-LRB-

-RRB-

E

h

eB

-LRB-

-RRB-

i

with

A

-LRB-

-RRB-

i

-LSB-

-LRB-

-LRB-

-RRB-

-RRB-

-LRB-

-LRB-

-RRB-

-RRB-

-RSB-

h

NR

-LRB-

-RRB-

NR

-LRB-

-RRB-

NR

-LRB-

-RRB-

i

and

B

-LRB-

-RRB-

i

p

n

-LRB-

-RRB-

i

-LRB-

-RRB-

i

-LRB-

-RRB-

i

-LRB-

-RRB-

i

p

n

-LRB-

-RRB-

i

-LRB-

-RRB-

i

-LRB-

-RRB-

i

-LRB-

-RRB-

O

-LRB-

-RRB-

The

term

B

-LRB-

-RRB-

contains

all

the

contribution

from

the

random

variables

and

and

is

of

the

form

i

-LRB-

-RRB-

this

leads

with

-LRB-

-RRB-

to

E

h

eB

-LRB-

-RRB-

i

exp

-LRB-

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

cubic

terms

in

O

pn

-RSB-

-RRB-

Let

us

define

G

-LRB-

t

-RRB-

F

-LRB-

-LRB-

t

-RRB-

-RRB-

and

R

-LRB-

-RRB-

NR

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

we

get

E

h

ei

-LRB-

-RRB-

i

exp

-LRB-

i

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RRB-

-LSB-

n?R

-LRB-

-RRB-

R

-LRB-

-RRB-

n?R

-LRB-

-RRB-

-RSB-

cubic

terms

in

O

pn

-RSB-

-RRB-

Now

we

remember

that

we

are

actually

interested

in

the

normalized

process

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-LRB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

nG

-LRB-

t

-RRB-

-RRB-

pn

Substituting

pn

for

and

pn

for

we

get

E

ei

-LRB-

-RRB-

exp

-LRB-

-LSB-

R

-LRB-

-RRB-

R

-LRB-

-RRB-

R

-LRB-

-RRB-

-RSB-

O

pn

-RSB-

-RRB-

which

proves

the

Theorem

Examples

Let

us

illustrate

our

technique

with

two

examples

In

Section

we

derived

the

following

results

for

the

model

with

unbounded

urns

-LRB-

-RRB-

G

-LRB-

t

-RRB-

F

-LRB-

-LRB-

t

-RRB-

-RRB-

-LRB-

-LRB-

t

-RRB-

ff

-RRB-

NR

-LRB-

-RRB-

ffe

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

ffe

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

-LRB-

-RRB-

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

This

gives

-LRB-

t

-RRB-

-LRB-

t

-RRB-

ff

-LRB-

see

-LRB-

-RRB-

-RRB-

A

weighted

process

with

insertions

and

deletions

For

a

sequence

of

insertions

and

deletions

starting

from

and

arriving

at

an

empty

relation

we

have

-LRB-

t

-RRB-

t

-LRB-

t

-RRB-

and

-LRB-

-RRB-

-LRB-

-RRB-

The

survival

probability

is

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Theorem

leads

to

E

-LSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-RSB-

nff

e?t

-LRB-

t

-RRB-

COV

-LRB-

-RRB-

ne?oe

ff

h

eo

ff

o

oeoe

ff

with

o

-LRB-

-RRB-

and

oe

-LRB-

-RRB-

-LRB-

-RRB-

A

unweighted

process

with

insertions

deletions

and

queries

We

study

now

a

sequence

of

insertions

deletions

and

queries

starting

from

an

empty

structure

and

arriving

at

a

structure

of

size

x

apn

We

are

still

assuming

that

the

urns

have

an

infinite

capacity

We

have

F

-LRB-

x

-RRB-

-LRB-

t

-RRB-

and

NR

as

above

but

the

functions

and

are

those

corresponding

to

P

-LRB-

t

-RRB-

xt

at

-LRB-

s

t

-RRB-

s

-LRB-

t

-RRB-

-LRB-

we

recall

that

x

pI

pD

-RRB-

and

and

are

as

follows

-LRB-

-RRB-

pD

x

-LRB-

-RRB-

x

pD

x

We

give

below

the

expectation

and

the

covariance

of

the

process

size

of

the

projection

in

this

case

E

-LSB-

S

-LRB-

-LSB-

nt

-RSB-

-RRB-

-RSB-

nff

e

xt

ff

p

n

ate

xt

ff

COV

-LRB-

-RRB-

n?R

-LRB-

-RRB-

with

R

-LRB-

-RRB-

e

x

-LRB-

-RRB-

ff

-LRB-

-RRB-

pD

x

ff

pD

x

-LRB-

-RRB-

Projection

maximum

We

have

shown

in

Section

that

Y

-LRB-

-LSB-

nt

-RSB-

-RRB-

nG

-LRB-

t

-RRB-

p

nV

-LRB-

t

-RRB-

O

-LRB-

-RRB-

with

G

a

deterministic

curve

and

V

a

gaussian

process

To

analyze

the

maximum

of

Y

-LRB-

-LSB-

nt

-RSB-

-RRB-

we

must

first

know

whether

G

-LRB-

t

-RRB-

has

a

maximum

for

t

-RSB-

-LSB-

If

G

-LRB-

t

-RRB-

is

maximized

at

t

then

it

is

easily

checked

that

Y

-LRB-

-LSB-

nt

-RSB-

-RRB-

is

also

maximized

at

t

So

assume

that

-LRB-

t

-RRB-

for

a

unique

t

-RSB-

-LSB-

-LRB-

local

maxima

can

be

analyzed

similarly

-RRB-

Set

G

G

-LRB-

t

-RRB-

and

-LRB-

t

-RRB-

It

is

equivalent

to

analyze

the

maximum

of

Y

or

of

the

normalized

process

obtained

by

a

change

of

scale

in

the

process

and

in

the

time

X

-LRB-

t

-RRB-

Y

-LRB-

-LSB-

nt

-RSB-

-RRB-

n

G

pn

V

-LRB-

t

-RRB-

p

n

-LRB-

G

-LRB-

t

-RRB-

G

-RRB-

-LRB-

-RRB-

We

shall

use

a

technique

based

on

the

results

of

Daniels

-LSB-

-RSB-

which

applies

precisely

to

a

process

of

the

form

-LRB-

-RRB-

and

which

we

recall

below

The

basic

results

Consider

a

Gaussian

process

V

-LRB-

t

-RRB-

superimposed

on

a

curve

y

-LRB-

t

-RRB-

If

we

look

for

its

maximum

m

max

-LSB-

V

-LRB-

t

-RRB-

y

-LRB-

t

-RRB-

-RSB-

and

for

the

time

t

at

which

this

maximum

occurs

it

is

equivalent

to

search

for

the

hitting

time

of

V

-LRB-

t

-RRB-

to

the

absorbing

boundary

m

y

-LRB-

t

-RRB-

It

is

well

known

-LRB-

see

Durbin

-LSB-

-RSB-

-RRB-

that

near

the

crossing

point

V

-LRB-

t

-RRB-

behaves

locally

like

a

Brownian

Motion

BM

or

a

variant

of

it

such

as

a

Brownian

Bridge

BB

It

is

also

known

that

the

hitting

time

and

place

densities

for

a

BB

can

be

deduced

from

the

hitting

time

density

for

a

BM

-LRB-

see

for

instance

Louchard

-LSB-

-RSB-

for

a

constant

boundary

and

Csaki

et

al.

-LSB-

-RSB-

for

a

general

proof

-RRB-

Assume

that

y

-LRB-

t

-RRB-

is

given

by

y

-LRB-

t

-RRB-

p

ny

-LRB-

t

-RRB-

n

-LRB-

-RRB-

and

that

it

has

a

unique

maximum

at

t

with

y

-LRB-

t

-RRB-

-LRB-

if

necessary

we

translate

the

origin

-RRB-

Daniels

and

Skyrme

-LSB-

-RSB-

have

computed

the

asymptotic

hitting

time

and

place

density

In

the

Gaussian

process

case

with

covariance

C

-LRB-

s

t

-RRB-

s

t

Daniels

-LSB-

-RSB-

has

matched

the

local

behaviour

of

C

-LRB-

s

t

-RRB-

with

the

BM

-LRB-

or

one

of

his

variants

-RRB-

covariance

near

t

In

the

BB

match

we

have

-LSB-

V

-LRB-

t

-RRB-

p

ny

-LRB-

t

-RRB-

-RSB-

p

A

-LSB-

BB

-LRB-

t

-RRB-

p

n

-LRB-

t

-RRB-

-RSB-

on

t

-LRB-

T

-RRB-

with

BB

-LRB-

T

-RRB-

y

-LRB-

t

-RRB-

-LRB-

t

-RRB-

and

A

is

some

constant

We

can

deduce

the

density

of

the

maximum

m

and

time

t

from

Daniels

-LSB-

-RSB-

-LRB-

-RRB-

and

Daniels

and

Skyrme

-LSB-

-RSB-

-LRB-

-RRB-

We

first

need

to

introduce

some

notations

Let

-LSB-

@sC

-LRB-

s

t

-RRB-

-RSB-

t

-LSB-

@tC

-LRB-

s

t

-RRB-

-RSB-

t

c

C

-LRB-

t

t

-RRB-

A

t

c

T

t

c

T

cA

-LRB-

-RRB-

B

-LRB-

t

-RRB-

u

-LRB-

t

t

-RRB-

-LRB-

-RRB-

Let

also

R

-LRB-

x

-RRB-

exp

-LRB-

-RRB-

H

-LRB-

x

-RRB-

with

H

-LRB-

x

-RRB-

Z

esx

ds

Ai

-LRB-

-RRB-

Ai

is

the

classical

Airy

function

Let

f

-LRB-

x

-RRB-

-LRB-

x

-RRB-

R

-LRB-

x

-RRB-

-LRB-

x

-RRB-

H

-LRB-

x

-RRB-

H

-LRB-

x

-RRB-

R

and

are

tabuled

in

Daniels

-LSB-

-RSB-

note

that

f

-LRB-

-RRB-

Finally

define

Z

-LSB-

R

-LRB-

x

-RRB-

x

-RSB-

dx

The

result

of

Daniels

and

Skyrme

is

as

follows

Theorem

The

random

variable

m

max

-LSB-

V

-LRB-

t

-RRB-

y

-LRB-

t

-RRB-

-RSB-

is

asymptotically

Gaussian

with

mean

E

-LRB-

m

-RRB-

n

-LRB-

-RRB-

and

variance

-LRB-

m

-RRB-

c

-LRB-

-RRB-

The

conditioned

maximum

mjt

is

asymptotically

Gaussian

with

mean

and

variance

E

-LRB-

mjt

-RRB-

n

-LSB-

-LRB-

u

-RRB-

-LRB-

u

-RRB-

-RSB-

-LRB-

-RRB-

-LSB-

mjt

-RSB-

c

-LRB-

-RRB-

The

joint

density

of

m

and

t

is

given

by

-LRB-

m

u

-RRB-

dmdu

s

-LRB-

-RRB-

-LRB-

f

-LRB-

u

-RRB-

n

-LRB-

u

-RRB-

A

c

-LRB-

u

-RRB-

R

-LRB-

u

-RRB-

c

R

-LRB-

u

-RRB-

-LRB-

u

-RRB-

c

O

-LRB-

n

-RRB-

o

dmdu

-LRB-

-RRB-

where

u

has

density

f

-LRB-

u

-RRB-

All

expectations

and

densities

have

relative

errors

of

order

O

-LRB-

n

-RRB-

Application

to

the

projection

size

From

Theorem

we

know

that

Y

-LRB-

-LSB-

nt

-RSB-

-RRB-

p

n

np

n

G

V

-LRB-

t

-RRB-

p

n

-LSB-

G

-LRB-

t

-RRB-

G

-RSB-

o

O

-LRB-

-RRB-

with

COV

-LSB-

V

-LRB-

-RRB-

V

-LRB-

-RRB-

-RSB-

R

-LRB-

-RRB-

The

O

-LRB-

-RRB-

term

is

non-uniform

in

V

but

it

is

easy

to

check

that

Max

-LRB-

Y

-RRB-

is

only

affected

by

a

O

-LRB-

-RRB-

term

Comparing

with

-LRB-

-RRB-

we

must

identify

y

-LRB-

t

-RRB-

with

G

-LRB-

t

-RRB-

G

We

can

now

compute

c

R

-LRB-

t

t

-RRB-

R

-LRB-

-RRB-

t

R

-LRB-

-RRB-

t

-LRB-

-RRB-

A

B

The

result

of

Daniels

leads

to

the

following

theorem

Theorem

M

max

-LSB-

-RSB-

Y

-LRB-

-LSB-

nt

-RSB-

-RRB-

n

G

m

p

n

O

-LRB-

-RRB-

where

m

is

a

random

variable

characterized

by

-LRB-

-RRB-

-LRB-

-RRB-

The

time

t

when

the

maximum

occurs

is

a

random

variable

characterized

by

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

The

constants

are

given

by

-LRB-

-RRB-

Let

us

again

illustrate

our

theorem

with

an

example

For

the

unbounded

urn

model

and

the

process

-LRB-

no

deletions

or

queries

-RRB-

the

function

G

-LRB-

t

-RRB-

is

maximized

for

t

so

G

-LRB-

e

-RRB-

and

B

e

c

ffe

ff

ff

A

Theorem

is

now

applicable

Acknowledgments

We

are

grateful

to

C.

Lefevre

for

several

discussions

on

birth

and

death

processes

References

-LSB-

-RSB-

BILLINSLEY

P.

Convergence

of

Probability

Measures

Wiley

-LSB-

-RSB-

CSAKI

E.

FOLDES

A

and

SALMINEN

P.

On

the

Joint

Distribution

of

the

Maximum

and

its

Location

for

a

Linear

Diffusion

Ann

Inst

Henri

Poincar?e

-LSB-

-RSB-

DANIELS

H.E

and

SKYRME

T.H.R.

The

Maximum

of

a

Random

Walk

whose

Mean

Path

has

a

Maximum

Adv.

Appl

Prob.

-LSB-

-RSB-

DANIELS

H.E.

The

maximum

of

a

Gaussian

Process

whose

mean

Path

has

a

Maximum

with

an

Application

to

the

Strength

of

Bundles

of

Fibres

Adv.

Appl

Prob.

-LSB-

-RSB-

DURBIN

J.

The

First-Passage

Density

of

a

Continuous

Gaussian

Process

to

a

General

Boundary

J.

Appl

Prob.

-LSB-

-RSB-

FLAJOLET

PH.

PUECH

C.

and

VUILLEMIN

J.

The

Analysis

of

Simple

List

Structures

Inform

Sci.

-LSB-

-RSB-

GARDY

D.

Normal

limiting

distribution

for

projection

and

semijoin

sizes

SIAM

Journ

on

Disc

Math.

-LRB-

-RRB-

-LSB-

-RSB-

JOHNSON

N.L.

and

KOTZ

S.

Urn

models

and

their

application

Wiley

Sons

-LSB-

-RSB-

KARLIN

S.

and

TAYLOR

H.M.

A

second

Course

in

Stochastic

Processes

Acad

Press

-LSB-

-RSB-

LOUCHARD

G.

Brownian

Motion

and

algorithms

complexity

B.I.T.

-LSB-

-RSB-

LOUCHARD

G.

Random

walks

Gaussian

processes

and

list

structures

Theor

Comp

Sc.

-LSB-

-RSB-

LOUCHARD

G.

SCHOTT

R.

and

RANDRIANARIMANANA

B.

Dynamic

algorithms

in

D.E.

Knuth

's

Model

a

Probabilistic

Analysis

Theor

Comp

Sc.

-LSB-

-RSB-

LOUCHARD

G.

Trie

size

in

a

dynamic

list

structure

In

CAAP

Lecture

Notes

in

Computer

Science

Springer

Verlag

Appendix

some

preliminary

results

for

Section

A

few

fundamental

formulae

provide

the

building

blocks

for

all

the

covariance

computations

of

Section

Let

-LRB-

-RRB-

and

-LRB-

-RRB-

We

recall

that

is

the

indicator

function

-LRB-<=-RRB-

I

-LRB-

-RRB-

that

Z

-LRB-

resp

Z

-RRB-

is

the

probability

that

the

urn

Ui

is

empty

at

time

-LRB-

resp

-RRB-

and

that

Z

i

j

is

the

joint

probability

that

the

urn

Ui

is

empty

at

time

and

that

the

urn

Uj

is

empty

at

time

We

shall

use

notations

such

as

Z

-LRB-

A

-RRB-

for

the

model

with

infinite

urns

and

Z

-LRB-

B

-RRB-

for

the

model

with

bounded

urns

Unbounded

urns

We

first

derive

the

probability

that

a

given

urn

Ui

is

empty

at

the

time

Z

-LRB-

A

-RRB-

d

Obviously

the

probability

that

Ui

is

empty

at

the

time

is

Z

-LRB-

A

-RRB-

d

Now

the

joint

probability

that

the

urn

Ui

is

empty

at

the

times

and

is

the

probability

that

the

urn

is

empty

at

the

time

and

that

at

the

time

of

the

balls

added

between

and

has

survived

Z

i

i

-LRB-

A

-RRB-

d

The

joint

probability

that

the

urn

Ui

is

empty

at

the

time

and

that

the

urn

Uj

is

empty

at

the

time

is

computed

by

considering

the

number

k

of

balls

present

in

the

urn

Uj

at

the

time

Z

i

j

-LRB-

A

-RRB-

d

X

k

k

d

k

d

-LRB-

-RRB-

k

d

d

As

C

i

j

Z

i

j

Z

C

i

j

-LRB-

A

-RRB-

-LRB-

-RRB-

-LRB-

d

-RRB-

-LRB-

d

-RRB-

Let

us

now

proceed

to

the

asymptotics

We

shall

use

the

equality

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

to

simplify

some

formulae

Z

-LRB-

A

-RRB-

-LRB-

-RRB-

ff

-LRB-

O

-LRB-

n

-RRB-

-RRB-

Z

i

i

-LRB-

A

-RRB-

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

-LRB-

O

-LRB-

n

-RRB-

-RRB-

Z

i

j

-LRB-

A

-RRB-

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

-LRB-

O

-LRB-

n

-RRB-

-RRB-

To

compute

the

asymptotic

value

of

C

i

j

we

introduce

the

function

h

-LRB-

x

-RRB-

-LRB-

x

-RRB-

C

i

j

-LRB-

A

-RRB-

d

h

-LRB-

d

-RRB-

h

-LRB-

-RRB-

d

-LRB-

-RRB-

d

and

finally

C

i

j

-LRB-

A

-RRB-

-LRB-

-RRB-

e

-LSB-

-LRB-

-RRB-

-LRB-

-RRB-

-RSB-

ff

Bounded

urns

From

Lemma

the

distribution

of

the

number

of

balls

in

a

given

urn

at

the

time

is

asymptotically

binomial

with

parameters

ffi

and

-LRB-

-RRB-

fi

hence

the

probability

that

an

urn

is

empty

at

the

time

is

Z

-LRB-

B

-RRB-

-LRB-

-RRB-

ffi

The

probability

that

an

urn

is

empty

at

the

time

is

similarly

-LRB-

B

-RRB-

-LRB-

-LRB-

-RRB-

fi

-RRB-

ffi

Now

the

probability

that

the

urn

Ui

is

empty

at

the

times

and

is

the

product

of

the

probability

that

Ui

is

empty

at

the

time

by

the

conditional

probability

that

Ui

is

empty

at

the

time

knowing

that

it

is

already

empty

at

the

time

Z

i

i

-LRB-

B

-RRB-

Z

-LRB-

B

-RRB-

-LRB-

-RRB-

The

expression

we

have

just

derived

for

Z

-LRB-

B

-RRB-

gives

Z

i

i

-LRB-

B

-RRB-

-LRB-

-RRB-

ffi

-LRB-

-RRB-

-LRB-

-RRB-

ffi

-LRB-

-RRB-

ffi

As

in

the

unbounded

case

the

joint

probability

that

the

urn

Ui

is

empty

at

the

time

and

that

the

urn

Uj

is

empty

at

the

time

is

computed

by

considering

the

number

k

of

balls

present

in

the

urn

Uj

at

the

time

The

probability

that

the

urn

Uj

is

empty

at

the

time

knowing

that

it

has

k

balls

at

the

time

is

-LRB-

k

-RRB-

the

probability

that

the

urn

Uj

has

k

balls

at

the

time

knowing

that

the

urn

Ui

is

empty

at

the

same

time

is

computed

by

Lemma

in

which

we

substitute

ffi

for

the

total

number

of

possible

places

P

r

-LRB-

kj

-RRB-

ffi

k

ffi

k

ffi

ffi?k

As

Z

i

j

-LRB-

B

-RRB-

P

k

Z

-LRB-

B

-RRB-

P

r

-LRB-

kj

-RRB-

-LRB-

k

-RRB-

we

finally

get

with

ffi

n

Z

i

j

-LRB-

B

-RRB-

-LRB-

-RRB-

ffi

ffi

X

k

ffi

k

-LRB-

-RRB-

k

-LRB-

-RRB-

ffi?k

-LRB-

k

-RRB-

Now

we

use

the

relation

-LRB-

k

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

ffi?k

to

simplify

the

expression

of

Z

i

j

-LRB-

B

-RRB-

Z

i

j

-LRB-

B

-RRB-

-LRB-

-RRB-

ffi

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-RRB-

ffi

We

next

turn

to

the

computation

of

C

i

j

Z

i

j

Z

Probabilistic

arguments

give

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-RRB-

which

we

plug

into

the

approximate

value

of

-LRB-

B

-RRB-

-LRB-

B

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-RRB-

ffi

The

approximate

values

we

have

just

computed

for

Z

-LRB-

B

-RRB-

-LRB-

B

-RRB-

and

Z

i

j

-LRB-

B

-RRB-

show

that

C

i

j

-LRB-

B

-RRB-

Z

-LRB-

B

-RRB-

-LRB-

h

-LRB-

-RRB-

h

-LRB-

-RRB-

-RRB-

for

a

function

h

-LRB-

x

-RRB-

-LRB-

-RRB-

x

-LRB-

-LRB-

-RRB-

x

-RRB-

ffi

Now

ffi

n

so

h

-LRB-

-RRB-

h

-LRB-

-RRB-

-LRB-

-RRB-

ffi

n

which

gives

with

ffffi

C

i

j

-LRB-

B

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

ffi

-LRB-

-RRB-

ffi

Finally

we

can

simplify

the

asymptotic

expression

for

Z

i

j

-LRB-

B

-RRB-

that

we

shall

use

to

compute

COV

-LRB-

-RRB-

according

to

Lemma

As

-LRB-

O

-LRB-

n

-RRB-

-RRB-

and

-LRB-

see

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-RRB-

-LRB-

-RRB-

which

gives

finally

Z

i

j

-LRB-

B

-RRB-

-LRB-

-RRB-

ffi

-LRB-

-RRB-

ffi

Contents

Introduction

Databases

and

urn

models

Projections

and

the

occupancy

problem

in

urn

models

Dynamic

models

Main

ideas

and

results

Notations

The

process

P

related

to

the

total

number

of

balls

Urn

models

-LRB-

A

and

B

-RRB-

preliminary

results

General

form

of

the

covariance

Lemma

Model

A

the

urns

are

of

unlimited

size

Model

B

the

urns

are

bounded

Distribution

of

the

number

of

balls

in

an

urn

Conditional

distribution

Computation

of

COV

-LRB-

-RRB-

for

a

non-random

static

structure

A

unbounded

urns

and

insertions

B

bounded

urns

and

insertions

Non-random

static

structure

Unbounded

urns

and

return

to

an

empty

structure

A.

Bounded

urn

and

queries

B.

Proof

of

Proposition

Random

Structure

The

perturbation

on

the

expectation

and

covariance

of

and

The

covariance

of

and

Examples

Projection

maximum

The

basic

results

Application

to

the

projection

size

Appendix

some

preliminary

results

for

Section

Unbounded

urns

Bounded

urns

derived relation
gaussian process
occupancy problem
relational database
urn model