NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

30 September 1952

1103 “21-5119 195S

Progress Report for July-Sept „ 1952

on

Applications of the Theory of Stochastic

Processes to the Study of Trajectories

(NBS Project 1103-21-5119)

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, reprinting, orrepi from the Office of the Director, N: which a report has been specifics

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

blted unless permission Is obtained In writing

on Is not needed, however, by an agency for

i of that particular report for Its own use.

This is a special progress report to the U.S. Naval

Ordnance Test Station which sponsors NBS Project 1103-21-5119,

Application of the theory of stochastic processes to the

study of trajectories.

J. H. Curtiss

Chief, National Applied

Mathematics Laboratories

PROGRESS REPORT FOR JULY-SEPT. 1952

ON

APPLICATION OF THE THEORY OF STOCHASTIC PROCESSES

TO THE STUDY OF TRAJECTORIES

(NBS Project 1103-21-5119)

I . Summary

This report contains a summary of the work done

during the quarter. Results of this work are briefly

stated. Technical reports written in connection with

this project are mentioned but are transmitted separately.

II . Discussion of work done during the quarter . The study of strongly continuous stochastic processes

was continued during this quarter. A more intuitive formulation of the theorem proven in NBS report 1665 was found.

Using a recent result [Sankhya 11, 1951] of Professor Mann

it can be stated in the following manner: A stochastic

process with independent increments which can be realised

in the space of continuous functions has normally distributed

increments. This reformulation has of course no theoretical

significance, it should however appeal more to people interested in applications since it avoids the use of the

formal definition of strong continuity. This formulation

should also be useful in explaining why the assumption of

normality of the increments can be made without hesitation

on physical grounds if one deals with trajectory data.

A characterization of the Wiener process was given by

proving the following theorem:

A stochastic process y(t) is a Wiener process 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(0) =0*

(to) The mean value function of the process is identically zero.

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

depends only on the length J* of the interval Over

which the increment is taken but is independent

of the location of the interval on the time axis.

-2 -

The question arose whether assumptions (c) is independent

of assumptions (a) and (b). This was answered by constructing processes with factorable variance function.

The following theorem was proved:

Let y(t) be a stochastic process, defined for t = 0

and denote by f(t,7') the variance of the increment

[y(t +7") - y ( t

)

]

• Assume that

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

and is strongly continuous in every finite

interval [a,b] where 0 # a<-b,

(ii) the mean value function of y(t) is identically

zero, i.e. Ey(t) = 0,

(iii) the initial value is zero, y(0) = 0,

(iv) f (t , T ) >0 for t = 0 and T >

0

(v) f (t, TO = h (t ) g ( T)

Then y(t) is a Gaussian process with covariance function

(3)

y(t

1 )y(t2

)

where t = min (t^

,

)

.

This theorem shows that there exist strongly continuous

processes with independent increments which are different

from the Wiener process. These processes could become

useful because they offer an alternative to the Wiener

process which is different from the well known Ornstein - Uhlenbeck process. However our study of the possible

application of the theory of stochastic processes to

trajectory data does not indicate at present any need

for such an alternative.

It is intended to condense the results obtained

in NBS reports 1665 and 1846 into a single paper on

strongly continuous stochastic processes and to submit

this paper to a technical journal for publication.

During the last visit of E. Lukacs and Professor

H. B. Mann at Inyokern problems connected with the

estimation of the mean value function of a Wiener

process were discussed extensively. It was agreed

that it would be desirable to demonstrate the procedure

by giving a complete numerical example. Some preliminary

work was done on constructing a suitable numerical example.