NATIONAL BUREAU OF STANDARDS REPORT

2201

ON STRONGLY CONTINUOUS STOCHASTIC PROCESSES

by

Eugene Lukacs

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

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NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-21-5119 January 15 ? 1953 2201

On strongly continuous stochastic processes*

by

Eugene Lukacs

Preprint

*The preparation o£ this paper was sponsored by the

U. So Naval Ordnance Test Station, Inyokern,

The publication, reprint

unless permission Is obt;

25, D, C, Such permlss

cally prepared If that a

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

part, Is prohibited

dards, Washington

t has been sped fit for Its own use.

On strongly continuous stochastic processes*

by

Eugene Lukacs

National Bureau of Standards, Washington, D.C.

1 . Introduction

In this paper we study strongly continuous stochastic

processes and give first a condition which assures that the

increments of a stochastic process y(t) a re normally distributed* This condition is of interest in connection with the

study of the Wiener process. This process affords a particularly simple model for certain phenomena because it has normally and independently distributed increments. It is shown

in the present paper that the normality of the increments

follows from a certain continuity property. Later we give a characterization of the Wiener process and discuss also another process with independent increments. We give next the

definitions which are necessary for the formulation of our

theorems

.

The inc remen

t

of a stochastic process y(t) over the time

interval (t, t + x ) is the random variable y(t + x) - y(t).

A stochastic process is said to be a process with independent increments if the increments _over non-overlapping

time invertals are completely independent of each other.

The preparation of this paper was sponsored by the U. S.

Naval Ordnance Test Station, Inyokern.
V- - ' : , / < :
2

A process y(t) is said to be strongly continuous [7] in the

closed interval [a,b] it to every e> 0 and ij > 0 there exists a

& = 6(e, jj) such that for every finite set S of points contained

in [a,bj

(1) P[ i£(S,e, S)j > 1 -

Here £(6,e,S) in the event that the inequalities |y(t^)-y(t, )j < e are simultaneously satisfied for ell pairs with

| t i-t

k! ^ & belonging to a finite set S of points contained in

[a,b]c The symbol P[ • . . J stands here and in the following for

the probability of the event [•••] in the brackets.

We shall write occasionally |y(6,£,S) for the event 4(6, £,S)

if it is necessary to point out that the event refers to the

process y(t). We are now in a position to formulate the condition mentioned in the introductory section.

Theorem 1 ; Let y(t) be a stochastic process and assume that

(i) y(t) is a process with independent increments)

(ii) y(t) is strongly continuous in the interval [a,bj.

Then y(b) - y(a) is normally distributed.

A theorem of this type is due to P. Levy [5] (theorem 18 , 3)»

However it might be of interest to give a different proof

using ane,6 condition for the definition of strdng continuity.

This definition is due to H. B. Mann [ 7 ].

2 a Khiratchine's theorem . For the proof we need a theorem of A. Khintchine which

gives conditions for the convergence to the normal distribution.

3

This theorem is given in a book by Khintchine [4j, published

in the Russian language. The following formulation was taken

from a paper by B. V, Gnedenko [3] which is available in an

English translation.

(n = ad inf) of random variables which are independent within each sequence* The random variables X are

said to be infinitesimal if for any e > 0 the relation

lira F[|X

ns | > e j = 0 holds uniformly in s (1 < s < k^).

I ’=£*00

We denote by F _ _ ( x ) the distribution function of the random

ns variable X „ and state

ns

Khintchine's theorem ; If the distributions of the suras

< 2 > T

n = X

n1 + Xn2--- + X

nk

n

of independent (within each sequence) infinitesimal random

variables X (1 < s < k ) converge to a limiting distribution,

0 5 O

then the necessary and sufficient condition for the limiting

distribution to be normal is that for any £ > 0

For the proof the reader is referred to Gnedenko's paper [3j.

3* Proof of theorem 1 We consider a sequence (S

n ) of subdivisions of the interval [a,bj. For the sake of simplicity we let S

n be the subn

(3)
. .

: • ... . '

. •

•

- •’ ...

-

'- -

; :>! : , :

*
4

-

division 5 ^ - \t n1 t, , 11,.,

u O t C- is we put t? = & + (b-a)j/n

J

, t”) of [a,b] into n equal

( j=Q,1 ,2, . . . 5 n.) . We v.' rite

parts, that

x . = y(t”) - v(t n .) for j = 1,2,,»..n and show first that n,j J J-i

the random variables x . are infinitesimal.

J By assumption (ii) the process y(t) is strongly continuous

in [ a ,b j . It is therefore possible to determine for every

e > 0 and vi > 0 a 6 = 6(e,^) such that for every subdivision

(b-a)

(4) P[ i(6,e,S

n

)] > I - n

We next choose a number N-N( e , jp such that N >

For any n > N the event ‘SCd^S ) implies that the n inequalIS ities I x„ • I < e (j=1,2,...n) are simultaneously satisfied.

• n , j

i - We conclude then from (4) that

(5) i - ft

l x

n ,j | < e 5 J = 1 , 2 , . . . n j and also

(6) i - n

< ?[

j Kj 1 < e J for j = 1 » 2 ,

.

• . n j if only

n > N . We clenote here by 21

|x‘n ? j 1 1. £ 5 j — 1 ^ cL

y © © o ^

IHt J the probability

that the:n inequalit;ies I x

1 n, ji * 6 (j = ) hold sisaTM

ultaneously (* )

T If R

A (i=1

,

2 ^ • © o ,n) are n events th.en P[R^ i-1 2 j o o <j ,n]

means the probability that all n events occur s imul taneously

.

Thus P[ ; i=1,2,...,n] > 1 - r\ means that the probability of

the simultaneous occurrence of all n events is at least equal

to 1 - q . This should be carefully distinguished from the statement P[R. j > 1- q (i=1 ,2, . • . ,n) which means that the probability

of the occurrence of each single event is at least equal to 1-q,

this statement does not imply anything about the probability

of the joint occurrence of the n events.
*

-

.

-£
5

For every e > 0 and ij > 0 it is therefore possible to find

an N=Ivi( e >rt ) such that for n >. N

(7) P[|x J >£ ^ f0r j = 1

> 2 >-"> n ‘

This shows that P[ |x n -j > e j converges (uniformly in j) to

zero as n —> oo , or in other words the % . are infinitesimal

J random variables. We consider the sequence of random variables

(8) x

n,n

We have shown that T is the sum of infinitesimal random

n variables which are by assumption independent (within each

sequence). Since

T

n = £*„,, ~ y(t j-1 )]= y(t“)-y(t^)=y(b)-y(a)

we see that the limiting distribution of the T

r is the distribution of the random variable y(b) - y(a).

We have already shown that it is possible to find for every

e > 0 and > 0 a N=N( £ ? rA ) such that

-S e j j ~ 1).*.?nj > 1 — rt

for n > N.

Since the increments x . are independently distributed

n } j

[by assumption ( i )

J

it follows that also

(9) P[ |x n J < e ; j = 1 ,2,...,k]

where e and ^ are arbitrary positive numbers and n >. N while

k is an integer less than n.
C .i»; -
6

For any subdivision 3

^ of [abj into n equal parts we introduce

the random variable

(10) U

abS

n = “®x C |x n,ll ' x

n ,2' ’•••’ l x n,n^*

The statement that the n inequalities j

x

n jj < e for j ~ 1,2,...n

hold simultaneously is equivalent to the statement M

abg

< e.

n Therefore

M

a bs — £ J ” ^ j xn i S — e 5 j “ 1 ? 2, • . . ,n] >1 - q

n

9 J and also

(ID P[Mabg

n

>

e j < n>

for all e > 0 and 0, provided that n > N.

We introduce the following n events?

' is the event that the inequality | x > e holds.

B.^ ' (for j = 2,3>**»?n) is the event that the j inequalities

•J

l x

n,ll ± £ >

l x n,2l ^ 6 l x

n, j-1 1 * £

> KJ > e hold

simultaneously. The events B

^ n

' , li.. “ '

, . .

.

,B

’*'’* are mutually

exclusive and exhaust all the cases for which M , ^ > e . Therea bo

n fore we see from (11)

(12) n

> P[M > £ j j - Z P[B

i (n)

] aD!, n j-1 •>

for all e,q> 0 provided that n > N.

Since the random variables x„ x„ „ n , 1

7 n , n independent we have for j = 2,...,n

are completely

7

P[Bj j = P[|xnjk J j< E ; k=1 ,2, —(j-1 ) j P[ x

njj > e]

It follows therefore from (9) that

(13) P[B“j > (1-

n) P[ lx J > ej (j =2,...,n)

J

* 1 n ? J

for any e > 0 and q > 0 if only n > N so that

(I*v) £ p[b"j > (1-n) Z Pi I x A > e i

j-i J ' j-i n,J

for any s > 0 and q

> 0 provided that o >. N.

From (12) and (14) we see that

n

(15) n /(i-n) > 2„xPL |x n>j | > e j

for any £> 0, q > 0 if n > N,

n

lim

n —> oo j

> • 1

Hence we conclude that

= 0 or

(16)

n £ f ®n,jW = 000 lxj>£

But (1 6 ) is exactly Khinchine's condition (3) so that we have

shown that the limiting distribution of the T , that is the

distribution of y(b) - y(a) is normal.

4. Some properties of strongly continuous processes . In this section we derive several lemmas.

Lemma 1 . If the process y(t) is strongly continuous in the interval [a,bj then it is continuous at every point of the interval, that means plim y(t +t) = y(t) if a < t < b and

z->0

a < t + t < b e Proofs We take for the set S the set consisting of the two

points t and t +t where t + t is an interior point of [a,bj.

d

The event ^(6 ,e,S) is then the event that fy( t+x)-y ( t)| < e for

|t| < S , The strong continuity of the y(t) process implies

that for every e > 0, and jj > 0 there exists a $ = 6 ( e, $|)

such that P (| y (

t-»- tt) - y(t) | > e ) > 1 - f. for |t[ < 6. By the

definition of convergence in probability this means that

plim [y(t + t) - y(t)j = 0

t-> 0

Lemma 2 . Let y ( t) be a stochastic process and denote by

m(t) = E y(t) its mean value function. Assume that

(i) y(t) is a process with independent increments

(ii) y(t) is strongly continuous in the interval [«,!>].

Then the mean value function m(t) is continuous in every point

of the closed interval [a,b].

Proof; Let t be a point of [a,bj and assume that t + t is an

interior point of [a,bj, The process y(t) is then strongly

continuous in the interval [t, t +t] and we see from theorem 1

that y ( t + t) - y( t) is normally distributed. Therefore the

random variable y(t + f ) - y(t) has only one median which is

identical with its mean m(t + t) - m(t). According to lemma 1 the random variable y(t + x ) - y(t) converges in probability

to the sure number zero, we can therefore conclude from a result of Slutsky [see for instance Frechet [2] pp 1 86-1 87 ] that

lira [m(t + t) - m(t)J = 0 which proves the lemma.

T-»0

Lemma 3 o Let y(t) be a stochastic process, defined for t > 0,

and denote by f(t,v) the variance of the increments y(t+T)-y(t)<>
- . "

-
9

Assume that

(i) y( t) is a process witSi independent increments

(ii) y(t) is strongly continuous in every interval [a,b]

(0 < a < b)

then the function f(t,x) is continuous in t.

Proofs We have clearly

y(t+T+X) - y(t) - [y(t + t + X ) - y(t +t)J + [y(t + '0-y(t)J.

On account of the assumed independence of the increments on

the right-hand side of this equation we have

(17) f(t,T+X)-f(t, t) = f(t + t, X).

We know from theorem 1 that the increment y(t+T + X)-y(t+ t)

is normally distributed. Lemma 1 shows that it converges in

probability to zero. Its variance f(t + t , X) must therefore

converge to zero as X goes to zero. We see then from (17)

that lim [f(t, i+ X) - f(t ? t)] = 0 so that lemma 3 is proven.

X-K)

f % \ 5* Characterization of the Wiener process v J

A process x(t) is said to be a Wiener process if

(i) it is a process with independent increments and

initial value x(0) = 0.

(ii) the increment x(t + i) - x(t) is normally distributed with mean zero and variance c t , (where

c > 0) o We are now ready to characterize the Wiener process.

Theorem 2 . A stochastic process y(t) is a Wiener process

(*) The Wiener process is also called fundamental random

process

.
r.

V

•

. •: *

'< - i

; :

v.;/ i>f!

:

- - .

9

-
10

if and only if the following three conditions are satisfied

(a) y( t) is a strongly continuous process with independent

increments and initial value y(Q) = 0.

(b) The mean value function of the process is identically

zero.

(c) The variance of the increments y(t + t

)

- y(i) depends

only on the length % of the interval over which the

increment is taken but is independent of the location

of the interval on the time axis*

Proofs The necessity of these conditions follows from well

known properties of the Wiener process (see [6j or [7J) so

that we have only to show that the conditions of theorem 2 are

sufficient* From (a) it is immediately seen that property (i)

of the Wiener process is satisfied* It follows from theorem 1

that the increments y(t + x) - y(t) are normally distributed while (b) shows that the mean of the distribution of the

increments is zero. We have therefore only to show that the

variance of the increment y(t + x ) - y(t) is ct . According to

condition (c) the function f(t,x) depends only on x so that

we may write f(t,x) = f(x), From (17) we see that f(x) satisfies the functional equation

(18) f ( x + h) = f(x) O f (h) *

Lemma 3 states that f (z) is a continuous function. It is well

known that the only continuous solution of ( 1 8) is f(x)=cx , Since f(x) is by definition a variance we see finally that

c > 0.
U. fj.. :??; v>:' i' ' -i • - ' ' :> I' /

"if:
6« Processes with factorable variance function of the increments.

In this section we give an example which shows that in a strongly continuous process with independent increments the

variance f(t,x) need not be independent of t.

Theorem 3 » Let y(t) be a stochastic process, defined for

t > 0 and denote by f (i,T) the variance of the increment

[y(t + t) - y(t)]„ Assume that the mean value function of

y(t) is identically zero and that the initial value is zero,

i.e. Ey(t) = 0 and y(0) = 0.

The process y(t) is a Gaussian process with mean value

function zero and covariance function

P eKt1 (19) cr / Y

-a, v = or 2, a

= c —*—

yCt^yltg) y(t)

eK _1

where t = min (t^,t

2

) and c > 0 if and only if

(i) y(t) is a process with independent increments and

is strongly continuous in every finite interval

[a,b] where G < a < b,

(ii) f (t, t ) > 0 for t >0 and t > 0

(iii) f(t, t) = h ( t) g( *t) We first prove the sufficiency of the conditions.

By means of theorem 1 we conclude from assumptions (i) and

from y(0) = 0 that the process is Gaussian, by assumption its

mean value function is identically zero. The purpose of assumption (ii) is to exclude some ,, improper,t processes. To see

this we assume that there exist values t and t > 0 such
i • ' - . . r . : / ' / ; - ; ./ f . r u ' . - -
12

that f(t

o

, v

Q

) =0. From the independence of the increments

and the additivity of the variance it follows then that

f ( t , t. ) = 0 if x 1 < and also f(t + x1 , x0 ) = 0 if

T

1 + t

2 - To*

'^ie increment8 y(^0+ ^

2 ).y(t + x^) have

therefore a degenerate distribution if ^ T

0

* conclude therefore from Ey(t) = 0 that y(t) = y(t ) if

t < t < t + x . Assumption (ii) excludes therefore processes which do not change over a fixed interval. We proceed

now to derive formula (19)* From assumption (iii) and from

(17) we see that

(20) h(t) [g( x + X) - g(x)] - h(t + t ) g(X)

Further we conclude from (ii) that h(t) / 0 for all t > 0

while g(X) ^ 0 for all X > 0. We may therefore rewrite (20)

as

(21 ) q( X + X ) - q(x) _ h( t + x ) g(X) h(t)

The left-hand side of this equation is independent of t but

depends on x and X, while the right-hand side of (21) is a function of x and t but is independent of X . This is only

possible if the quotient on either side is a function only of

x* We denote this function by P(x) and obtain

(22) h(t +x ) = P (x) h(t).

If we set t = 0 we have h(x) = h(0) P(x) hence

h(t + x ) - h(0) P(t +x ). Substituting these expressions

into (22) we obtain

13

h(t +t ) = is (0) p(t + t) = P(t) h(t) = P (t) h(0) P(t)

or

(23) P(t + t) » p(t) P(t) .

Since by lemma 3 g(x) is a continuous function of t the same is

true for P(t) so that the only solution of (23) is

(2*f) p(t) = e

Kt

and therefore

(25) h(t) = h(0) eKt

From (21) and (24) we obtain the functional equation for g(x)

( 26 ) g(x+X) = g(x) + eKT

g(X)

In case K = 0, this reduces to

( 26a) g(x+X) = g(x) + g(X)

and the process y(t) is then a Wiener random process. #e

f mpy therefore assume in the following that K ^ 0. We obtain

/; easily from (26)

«£•. &( t.. + tp)

g(t

1 + t 2

... +t

n

) = g(t

1 ) + e**! g(t2

) + e * g(t^) + ...

K( t

1 + tp+* • »+t .. ) + « 9(t„).

If in particular t i = t0 =• . .= t = t we have

i cL n n-1 v Knt 1 (27) g(nt) = g(t) JL * = 9(0 ~Vt~~ s=b eKt -1

Setting here t = ^

we obtain for integer n

K/n

g(£> = g(D e _1

eK -1

m and n

(28) g(E> = ,(D ^

In th© same manner we obtain for integers

mK/'n -1 e -1

From the continuity of the function g(\) we conclude finally

that for any real X

(29) gOO = g(l )

^ — (if K ji 0).

e* -1

The variance f(t,T) of the increment y(t +t) - y(t) is now

immediately obtained from assumption (iii) and from (25) and

(29) and is

(30) f (t

,

t) = c eKt e

K1C -1

eK -1

where we wrote for brevity c = h(0) g(1) = f(0 ? 1). By assumption y(0) = 0 hence y(t) = y(t) - y(0) so that the variance

of y(t) is obtained by substituting t = 0 and x= t into ( 30 ) 2 e** -1 hence = e-g- • The formula (19) for the covariance

e -1

function follows then easily from the independence of the increments.

We proceed to prove that the conditions are necessary. We

assume therefore that y(t) is a Gaussian process with mean

value function identically zero and initial value zero and

variance function given by (19) and show that (i), (ii) and

(iii) hold.

By an elementary computation we obtain from (19)

f ( t jT

)

= E[y(t+T)-y(t)] 2 = c eKt —— so that (ii) and

e K -1

(iii) are satisfied
v.

;

• :

jr -'m ;• •

' m •- •
15

Let now t,j,t 2

j t^,t^ be four real numbers such that

0 < < tg < 13 < then the intervals [t^tgj and

are nonoverlapping. Using (19) we see easily that

E[y(t2 )-y(t

1 )] [yCt^J-yCt^)] = 0.

The increments over nonoverlapping intervals are therefore

uncorrelated; since the process is Gaussian they are also independent.

We complete the proof by showing that the process is

strongly continuous. For this we use an idea of J. L. Doob [ 1 j

We write

Jit 1 (31a) V(t) = c ~—

eK -1

and

(31b) W(t) = ~ In [1 + (e

K

-1 ) t]

then

V[W(t)] = ct .

We define a new process £(t‘) by setting

(32) £(t’) = y ( t

)

where t = W(t') or t

=

V

The function t = W(t’) maps nonoveriapping intervals onto

nonoverlapping intervals, it follows therefore from the assumptions about the y(t) process that £(t*) is a process

with independently and normally distributed increments and

mean value function zero. Moreover it is easily seen that

g (0) = 0 and that E£(t‘) £(t* + z') = c t‘ for t* > 0

so that Var [ £ (

t

1 + t ’ ) - £( t ‘ ) 3= ct’. It follows then
v- 5> •- > ::
16

frora the definition given at the beginning of section 5 that

(32) transforms the process y(t) into a Wiener process. (In

case K < 0, y(t) is transformed into a segment of a Wiener

process). A t-interval [a,b] and « set S of points t of

[a,b] are mapped by (32) on a t'-interval and a set S' of

points . Since V(t) is uniformly continuous in any

closed interval [a,b] it is possible to determine for a given

6

1 > 0 a 6 = 5(6') such that the event $*,(6 1 >

‘ ) implies

the event % (o,e,S)

.

Since (theorem 2) the process |(t’)

is strongly continuous it follows that y(t) is also strongly

continuous

.
'

.. ; I K

:

! -

- <

-

-

- ;

.

• 1 , "•

.. i‘

! t; >

;

: ; i.

•

' h
17

7 * REFERENCES

[ 1 ] J. L. Doob, The elementary Gaussian processes. Annals

Math. Stat. Ip, 229-262 (19*4).

[ 2 ] M. Frechet, Gen^raiites sur les Probabiiit/s

.

aleatoires. 2nd ed» Gauthiers-Villars , Paris

^l/ments

1950 .

[ 3 j B. V. Gnedenko, Limit Theorems for Sums of Independent

Random Variables. American Mathematical Society Translation No. 45, New York (1951). Translated from IJspehi

Matem. Nauk 10, 115-165 (1944).

C 4 J A. Khintchine, Limiting distributions for sums of independent random variables, (in Russian) Moscow-Leningrad (1936).

[5] Paul Levy, Processus s tochas tiques et mouvement Brownien.

Gauthiers-Villars, Paris 1948.

[6] H. B. Mann, Estimation of parameters in certain stochastic processes. Sankhya vol. 11, 97-106 (1951).

[7] H. B. Mann, Introduction to the theory of stochastic processes cepending on a continuous parameter. In press as

No>t 24 in the Applied Mathematics Series, National Bureau

of Standards, Published, by the United States Government'- . Printing Office, Washington, D,C, (1953),

/
, • . . . * ' -
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functio'ns of the National Bureau of Standards are set forth in the Act

of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950.

These include the development and maintenance of the national standards of

measurement and the provision of means and methods for making measurements

consistent with these standards; the determination of physical constants and

properties of materials; the development of methods and instruments for testing

materials, devices, and structures; advisory services to Government Agencies on

scientific and technical problems; invention and development of devices to serve

special needs of the Government; and the development of standard practices,

codes, and specifications. The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services

and various consultation and information services. A major portion of the

Bureau’s work is performed for other Government Agencies, particularly the

Department of Defense and the Atomic Energy Commission. The scope of activities is suggested by the listing of divisions and sections on the inside of the

front cover. Reports and Publications

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the Government Printing Office: The Journal of Research, which presents complete papers reporting technical investigations; the Technical News Bulletin, which

presents summary and preliminary reports on work in progress; and Basic Radio

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to use for radio communications throughout the world. There are also five series

of nonperiodical publications: The Applied Mathematics Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous Publications.

Information on the Bureau’s publications can be found in NBS Circular 460,

Publications of the National Bureau of Standards ($1.00). Information on cali- bration services and fees can be found in NBS Circular 483, Testing by the National

Bureau of Standards (25 cents). Both are available from the Government Printing Office. Inquiries regarding the Bureau’s reports and publications should be

addressed to the Office of Scientific Publications, National Bureau of Standards,

Washington 25, D. C.