Section 11.3 FILE COPY

Do not remove NATIONAL BUREAU OF STANDARDS REPORT

NBS Project NBS Report

1103-21-511© $ p r i i 15S8

PROGRESS REPORT FOR JANUARY - MARCH 1952

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APPUECAT201 OF THE THEORY OF STOCHASTIC

PROCESSES T© THE STUDY OF TRAJECTORIES

(MBS Ps>©j©€t 1103-21-5119;

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

This report is issued

production in any font writing is obtained fr Washington 25, D. C.

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

on, reprinting, or re- ed unless permission in

l Bureau of Standards,
w

s
PROGRESS REPORT FOR JANUARY - MARCH 1952

on

APPLICATION OF THE THEORY OF STOCHASTIC

PROCESSES TO THE STUDY OF TRAJECTORIES

CMBS Projest 1 103-21-51 19}

lo SUMMARY

Tills report contains a summary of the background of the

above project, and a statement of the problems involved 0 The

work' done during the quarter is discussed and the results

are. stated,, A technical report entitled "A property of strongly

continuous processes” by Eugene Lukacs will be submitted

separately.

2

• .PXSCUSSION OF PROBLEM AND OF WORK DONE DURING THE QMET1R *

Ip, this new project an attempt is mad® to investigate

whether mathematical -statist!cal tools associated with the

theory, of stochastic processes can b© profitably applied to.

the analysis of trajectory data gathered at the Naval Ordnance.

Tes;t; Station j, Inyokern. California. The theory of stochastic

processes deals with time-dependent phenomena in which there,

is, -a probability relationship feerween the state at a given

Instant and one or more states at proceeding instants,* Some

of the;: difficulties encountered in analysing ordnance data

can fee- overcome by considering a trajectory to be a stochastic

process 6 . This approach eliminates the difficulties created

"by the fact that only one observation is available for the

position of the missile at ©ach instant 0 The first step in this work is the adoption of a stochastic

aiodelc As always in such a situation one has to find a

compromise between the wish t© establish a realistic model and

the necessity to us© a model which does not lead to a forbiddingly

complicated mathematical analysis,* The fundamental random

process {Wiener process} is a very simple process and methods
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5X19 PROG EFT' JAEf-SIAR 52

16 April 1952

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for the estimation of its parameters were developed under

a previous Bureau project C^leaentary Theory of Stochastic

Processesj Task 1103-11-1107/49-3) • This is the main reason

why it: seems desirable to attempt to use in this study the

fundamental random process as the stochastic model* Clearly^

this involves a certain amount of idealisation., It is therefore

desirable to safeguard against too radical departure fro®

reality by testing first whether the observed trajectory

data can be assumed to come from a fundamental random process*

Since no such test is known at present the first task

under this project is the development of a. test for the

hypothesis' that a sequence of observations comes from a

fundamental random process*

This necessitates a study of the various properties

of the fundamental random process in order to see which of

these properties should be emphasised in the test to be

developed*

The following theorem was derived in this connection

mainly by using a result of Khinehines

Let y{t> to© a stochastic process and assume that

Ci> yCi) is a process with independent increments*

Cii) jjr(t) is strongly continuous in the interval [a e bj*

Then yib) - y<a) is normally distributed*

This theorem shows that the normality of the increment©

follows fro® a certain continuity property of the process*

In view of the physical situation encountered in the study

of trajectories a continuity assumption is certainly reasonable* Therefore the test to be developed should not emphasise

the normality of the increments g the independence ©f the

increments seesas to be the property deserving greater

attention*
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5119 PBGG IF? JAM-MAR 52

18 April 1952

e=* ^ ess

A technical report containing the proof of this theorem

was completed and- will be isened at the beginning of the

next quayter 0

£, Lnk&cs visited Professor H 0 ®» Mann in February 1952 e and

discussed with him variolas aspects ©£ this project 0
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