302 DOC.

30

QUASIPERIODIC

PROCESSES

Doc.

30

A

Method for the Statistical

Use

of Observations of

Apparently

Irregular,

Quasiperiodic

Processes

A.

Einstein

[Manuscript][2]

[after

28

February 1914][1]

Suppose

that

one

observes

a

quasiperiodically fluctuating quantity

F

as a

function of

an

independent

t

for

a

very large

t-interval

T.

How

can one

obtain statistical data of

a

perspicuous

character

concerning

F from the observation? In what follows

I

present

a new

kind of method

by

which

to

attain this

goal,

in

the

hope

that it will

prove

applicable

in

practice.

Let

us

imagine

that F

is

expanded

as a

Fourier series for the entire interval

T,

so

that

one

has

F(0

=

+

£

An°°S 2jTn^ +

V/i

Z

1

/

...(1)

V

T

Of

course,

such

an

expansion

is

not

realizable,

and

even

if

it

were

to

be

realized,

it

would be of little

help.

For the

An

and

pn

would

vary

with

n

in

a

most

irregular

and

complex manner.

It is

therefore advisable

to

introduce the

following concept

familiar from the

theory

of radiation. Let

us

form the

mean

square

of those

2

quantities

An

(An2),

which

belong

to

a

region

An of

n

that

is

characterized

by

the

two conditions[3]

An

is

large compared

with

1

An/n

is

small

compared

with

1.

This

quantity

A2n,

which

is

analogous

to

the

intensity

in

the

theory

of

radiation,

yields-expanded

as a

function of

n-a

very complete

statistical

expression

for the

character

of the

changes

in

the

quantity

F.

Hence,

our

task

must

be

to

look for

a

method of

obtaining

this

quantity

that

does

not

require

the determination of the

individual

An.

To this end

we

introduce

a quantity

x(A),

which

we

call the "characteristic"

and which shall be defined

as

follows

T

X(A)

=

F(t)F(t +

A)

=

I

f

F(t)F(t + A)dt.

...(2)

0

Here

A

denotes

a quantity

that

is

small

compared

with

T.

It

is

easy

to

see

that this