Foundations of the theory of probability

§

5.Differentiationand

Integration

of

MathematicalExpectations

45

matical expectation E[x'(t)] exists (Property VII of mathe-

maticalexpectation,in

§

2). Letuschoosea fixed tanddenote

byA theevent

xjt

+

h)

• xjt)

h

x'(t)

>

£

TheprobabilityP

(

A)tendstozeroash

—

foreverye

>

0.Since

x{t+h)

• %{t)

M, x(t)\^M

holdseverywhere,

andmoreoverinthecaseA

\

xjt

+

h)-

xjt)

then

h

-At)

Ex(t+h)^-Ex(t)

_

Ex

,

{t)

xit+h)

• xit)

-x\t)

P(A)E

2

xit+

h)

-xit)

x'it) P{A)E

J

h

xit+

h)

• 
  

xit)

x\t)

^2M?iA)+

a.

Wemay choose thee

> arbitrarily, and P(A) is arbitrarily

smallforanysufficiently

smallh.Therefore

dt

Exit)

=lim

. Exit+h)-Exit)

Exit),

h+

whichwas
tobeproved.

Proofof TheoremII.Let

k

=

n

s

n

={]?x(t

+

kh),

^-~r-

b

Since S

n

converges to J

—

J

x(t) dt, we canchoose for any

a

e

> anN suchthatfromn^Ntherefollows theinequality

P(^)=P{|S,

-/|>£}<

£.

Ifweset

k=n

S:

=

l^Exit+kh)

=

EiS

n),

k=\

then

|S*-E(/)|

=

|E(S

W

-/)|^E|S

W

-/|

P(^)E

A

\S

n

• 
  

J\

+9(A)Ei|S

n

• 
  

J\{

^

2KP{A)

+

e

^

(2K

+

l)e.