NATIONAL BUREAU OF STANDARDS REPORT

1665

A PROPERTY 01 STRONGLY CONTINUOUS PROCESSES

by

Eugene Lu&acs

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL OUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Acting Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following

listing of the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its

title. A brief description of the activities, and of the resultant reports and publications, appears on the inside of the back cover of this report.

1. Electricity. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Electrochemistry.

2. Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic Technology. Length. Gage.

3. Heat and Power. Temperature Measurements. Thermodynamics. Cryogenics. Engines and Lubrication. Engine Fuels.

4. Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry.

Physical Electronics. Electron Physics. Atomic Physics. Neutron Measurements.

Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation.

Radiological Equipment. Atomic Energy Commission Instruments Branch.

5. Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical

Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

6. Mechanics. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters.

7. Organic and Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing

and Specifications. Organic Plastics. Dental Research.

8. Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy.

Corrosion.

9. Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals.

Building Stone. Concreting Materials. Constitution and Microstructure. Chemistry

of Mineral Products.

10. Building Technology. Structural Engineering. Fire Protection. Heating and Air

Conditioning. Exterior and Interior Coverings. Codes and Specifications.

11. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering.

Machine Development.

12. Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic Instrumentation.

13. Ordnance Development. Mechanical Research and Development. Electromechanical Fuzes. Technical Services. Missile Fuzing Research. Missile Fuzing Development.

Projectile Fuzes. Ordnance Components. Ordnance Tests. Ordnance Research.

14. Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular

Propagation Services. Frequency Utilization Research. Tropospheric Propagation

Research. High Frequency Standards. Microwave Standards.

15. Missile Development. Missile Engineering. Missile Dynamics. Missile Intelligence.

Missile Instrumentation. Technical Services. Combustion.
NBS PROJECT Key 14, 1952 NBS REPORT

1103-21-5119 1665

A PROPERTY OF STRONGLY CONTINUOUS PROCESSES*

The preparation of this paper was sponsored, hy the U. S e Naval

Ordnance Test Station, Xnyokera

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

This report is issu< in any form, either i from the Office of tl

ation, reprinting, or reproduction

permission in writing is obtained

, Washington 25, D. C.
\ «
A PHDPEBfT OP SmiQllW COSVXHOOtJS PROCESSES

• by

'Mgmm 2sulea©®

national Bureau of Standards

10 Introduction

5?h® purpose ©f this repeat Is 4© give a condition which assures that

the inerelfeeRts of a stochastic process y{%) are normally distributed*

This eo?iditi©s 1® of interest in connection with the study of the fun&amen~

tal random process* The fundamental random process - also called the

Wiener process. - afford® a particularly simple model for certain phenomena*

One of the properties of the fundamental random process is that its

increments are normally end independently distributed (for. a definition

m® [ 4 ] or [5 ] ) 0 in the present report it is ®how that the normality

of the Increments follows, fro® a certain continuity property*

20 Definitions and statement of the theorem,.

In this section w® give the definitions necessary to formulate the

theorems,

fh© increment of a stochastic process y(t-) oyer the time interval

(t 0 % vV) Is the random variahl® y('t ^T*)=y(t)o

A stochastic process is said to ho a process with independent

increments if the increment® over nea-overlappir,g t5n© intervals are

completely independent of essfe other*
\

.

'

-

*

.
A proses® y(t) is said to %& atgqy^ly oontianous iri .the

i

interval [ a,, b] if te every £>0 and^>0 there exists a. (g 0 saoh that for mezj' finite set - 8 of points ©ontaiaed in t £c"^ J U) P Ctu,t, S)] -S 1 -jp<

Bara ${<>,£ ,8) .Is tlM avast tbst tba IwwUUm | )| are siaail ianeonsly satisfied for all pairs (tj0 tj. } with- j % * « ^

‘©©longing- to & finite set So* points contained in [ &.,b]

The symbol ?[„* 0] gtfflds here and in the following for the probability

of ’tho ©v®at£ o o o j in the bracket®

’sle are now in a position to formulate the condition aenticneil in the

introtinetosy ©«@ tic

c

fhffeggf t X=®t y{ t) he a stochastic process and assosae that

-(1 ) wi t) it a process with independent i&eresaeats

CU ) yf t-) is strongly eontianono in the interval [a0 b ] „ Thm y(h)- yfo) is normally distributed*,

A theorem of this type 1® da© to PB Levy [3] { theorem 16 „ 3), However

it might he of interest to give a different proof using the notation

and te;m|3aology of Ho Be Mann!i s monograph [5] »

3c Khiatchin© tt s theoremFor tlx© proof we need a theorem of A- Xhlntchln® whi©h give* a condition®

for the ©om ergeno® to the normal distribution*, This theorem ic given i®

a hook kr B^4at«;hia«|5Whlished to the Hoesian language. 3:lu following

foranlatlon was taken from a paper by B« ?» Snedes&o llj whioh is avail"

able in art English tranelatiea*

1 ¥a ©onslder a aesjuence of sequences ***R

4n»la 2 ore, ad inf)

of random variables whisk are independent within ©*®h fesquenseo The

random variables X^.ar® said to be inftoftesimal if for •- • ^ - 'the

f relation Xisr 51 > I ] * 0 nnifomly in (.i % 0
*

:S

*

*
W@ denote by f^x) th® distribution function of th® random variable

aad ©tat® ghijmfcehine l' ® tfaeorm?. If the distribution© of the suae

*£n

s + X„ 0 * 6oo 4* X nl n? nkn

@f independent ( within @mh mqa®m@) infinitesiaal ran&osa variable©

xnl"( 1® ^ ^ kn ) converge to a limiting di®tribution0 then the

necessary and ©efficient condition for the limiting distribution to be

aorraal is that for any € ^ 0

( 3 ) 11m

n~$a> i ^F^ix) « 0 . o f”1 \%\ik

For th© proof the reader 4® referred t© (H$s&enko D s paper £ 1 ]J 0 40 Proof of the theorem stated in section 2

©onsidor a sequence |sa | ©f subdivisions of the internal [a 8 b]

For the ©ak© of simplicity w© let Sn b© the subdivision

S

a

s C^o® ^1 G °°°o *5 ) of [a s b] into n equal parts,, that Is w© pat

a

a + ®Oc l 0 20 ° ° O 0 n ji

A

*to 0 J

83

^ ~ £©g> J® 1 0 30 o,o 0 B and ©how first that the

randon variables ss^ j are infinitesimal.

By asswptlo® (ii) the process y(t) is strongly continuous in [a s b] ,

It ip therefor© possible to deteraain© for every 6 >0 @M l|>0 a rjj)

*tlt

)

such that for ©very subdivision W P [£'id,tt %)] 4 i -q

W@ d©n©t© by p[| j |g g g Ja

the a inequalities I ^ 1 0 Sr

,2. ra] the probability that

, a) hold siwiltaneously f>)

855 Ip §q a o u _ u _ ar© n ©vents then P [E^g i® l 0 3c n] mesas the

^

V

probability that al-lR tv©nts occur simltaneouslyo Thu© P[ Ejg i® XpSB* ou n^l«te

mean© that the probability ©£ the simultaneous occurrence of ill n events 1© at |

l@a>it equal t@ 1 - n 9 This should bs carefully distinguished fro® the ©tat©- 7

®©at P [%] » I*, nfa m 1 0 20 ooo w which means that toe probability off the

y ©@©urr©®c© of ©ash singl© <sv®nt is at least eqjaal to this statement do©®--’

not imply anything about the probability of the Joint occurrence of the a ©vents
*

-

.
W© a©st choose a m&a^sr’HsslC £ p n

)

such that N &

For any np N the event' u (e, 1 8 Sn) is®pXi©® that th©

„ \: ^ | ^£(^2. 2 co d) are einralianeouely satisfied,, We ©onelude then

fro® i it.)- 'that • .

(6) 1 * n “ PC t ^.^1 « rf] and also

(6) 1 - rj » P[|x

n

, 1 ® 4] for j ~ l p 2 e oon a If only n « No

( b-a

;

„

inoqualitiee

For ©veryl^G and i|<^0 it i© therofar© possible t@ find an

N nN (f,,i| ) s«©h that for n a N

C?) P[ I xii 0 j

I 3 ^ ^

for j s l,2,.oon 0

This ©hove that P [ [ xn j>4] ©onverges (mnifersaly in J) to *ero m

n *£-OD 0 or in other word© th©

j

&r© infinitesimal random variables

W© ©on®l&©r the sequence of random variables

( 8 ) Tn

m x

n^

x

n^ 2

+ i'°° * *n,n

W© have shot® that S?E

is the mm ©f infis.it esiieaX random variable®

whleh are by assumption lss&®p€!»&©Bt (within @a@h sequent©) s Sis©®

r„. £ ^ , « £ 0(0 - y(t" ,)3 » y(t£)-jr ( t“)* y(b)-yU)

We ©e© that the limiting distribution of the ®

R

is the distributtorn ©f the

random variable yfb) - y{a)o

We foav© already shown that it is possible to find, far ®w@sj£?0 .and r^>Oja

H®M(g s i|) m©h that P[jx,

j

1 “ f l J ® l 5 ooogH] ^ 1

for si ® N0

Slnoe the is$sr©s>i©nt© are iad@p©32d®ntly distributed [by assumption (JL)]

it follows that -@1@®

( 9 ) PtlaCn^jl « tl J ® 1,2,000k] » 1«V

where 4. and j| are arbitrary positive numbers and R® S while te is an

integer 1®©® than a: ,

For w subdivision of Cab] into n equal parts ve introduce the random

variable . . , . .

' _ -

( 1°) *fcbsn « C|xns ii JXjj. 2 j, «*,o*o

6 (Kjj jjllo- ' '

ffe© statement that th® a inequalities f 3^ ^ <£fQP J ^l s 2 ?00 n

hold simultaneously is equivalent to the statement Ma

^

« Co

fheref@r® l

• "

• pt\bsn

*•& 3 - PClxn

.

ej J -1 , 2 ,. ..n] # 1-n

and also

(U) p^ Ma’bS21 ^ Cl *

*| 9

f@r all t > 0 sadrs^O ^ provided that n # Ho

W@ intro&m©© th© following a ©vsntsg

_ (n)

B

1 ®^©at that th© inequality |x J >t holds,'

Wf)X

(a)

* (for J B 2»30 o»4) is th® ©y@at that th® J in©qmalltl@®

I *0,3.1

s £» l*nt2

l * < » ««»”l x

B9 j=iig f»l*

n ,l>£ hold

»taaX4ara®oasly0 a® wants B, ‘ )

^

g

C“)

X 0 3 ooooo 'a &r® mutually exclusive and exhaust all th© cases for which M . / > £ D fh©r©f@r@ w® s©@ fro® (n) / '

(is) - PCHabs„H3 s £ P[B C

, n)

] . for aiU? 0 and lp©0 provided that n ^ Ho

Sias© th© rando® variables « o o ogStji^ji ar@ ©®®pl®t@Iy independent

w© hav© f@r J«2,. OB n

PC®"] Ptlx^jjl* t

%

to>l,2,.. 0 (J-l)] PClxn ,|>£3*

It follow© th©r©fa>r© tmm (8>) that "

(13) P[B^] i (i=i|) PClXnjOC] (J-2,...,n)

for &wi? 0 and ri> Oif only ni N s@ that

(u) i, PCB“ 3 6 U->}) £ PC l*„ ,l> t ]

for

>1 J * J©1

^

any L >0 and r| P Oprovldsd that n N 0 Fro® (12) and (14) v;e s®a that

/M

>
JC

'
6=

« l5) yu-*}) * £

pc!x

nsJ l>n ,

fox* any£»>Q e i^>0 If- niN« Hence w@ oomlnd® that

11m 2L pC|x J>g] « 0 or

n §> m jssi

n p (is) lim > /

dP (x) * o

n-=p coo c « jx ^9 « J 1 !x|«

But (16) 1 b ®52©e tly SJain©Ma@ u s ©oa&itiou ( 3) ©@ that m k&w® shorn that

the limiting distribution of the T # that ia the distribution of y (b) - y(&)

is normal

/ %- vrenGmu,

Be ¥o Gnedenkos Mralt Theorem® for Sots ©£ Independent Bandogs

Variables* American. Mathematical S@stetj Translation 1©0 460 Hew ¥©g>k (!95X) 0 Translated fro® Bspehi Mate®0 Hss&k 100 115-165

(1944) 0 A0 .lfaiatQhln® n Limiting distribution® for sots @f independent

random variables<> ( in Russian) Moeoovr «= Leningrad (1958) „

Paul L@wy Pro©©®®©© steehaetlques ®t oowement 3y@®lsao

Gauthiers ¥i!X©,r®0 Paris 1948

Ho Bo Mann Estimation of parameters In ©ertain stochastic prosssseso

Sankhya ?oL 11 0 .97»106 0 (1951)

'Ho So Mann Introduction t© the theorj of stochastic processes

depending @s& a continuous parara®t©g, 0

HoBoSo report #1893 (1951) t© Is© published a® AMS S4„
'

'
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions 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 fisting of divisions and sections on the inside of the

front cover. Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and

devices or published papers and reports. Reports are issued to the sponsoring

agency of a particular project or program. Published papers appear either in the

Bureau’s own series of publications or in the journals of professional and scientific

societies. The Bureau itself publishes three monthly periodicals, available from

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

Propagation Predictions, which provides data for determining the best frequencies

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.