PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 7 SEPTEMBER 15, 1921 Number 9 
THE AVERAGE OF AN ANALYTIC FUNCTIONAL 1 
By Norbert Wiener 
Department op Mathematics, Massachusetts Institute op Technology 
Communicated by A. G. Webster, March 22, 1921 
1. Conceive a particle free to wander along the #-axis. Suppose the 
probability that it wander a given distance independent 
(1) of the position from which it starts to wander, 
(2) of the time when it starts to wander, 
(3) of the direction in which it wanders. 
It may be shown 2 that under these circumstances, the probability 
that after a time, t, it has wandered from the origin to a position lying 
between x = x 0 and x=X\ is 
1 r xx — — 
-= I e ct dx 
where t is the time and c is a certain constant which we can reduce to 1 
by a proper choice of units. This choice we shall make in what follows. 
The exponential form of this integral needs no comment, while the mode 
in which / enters results from the fact that 
l r xi _ * 2 l A 30 r i r xi _ (y-*) 2 "1 — - 
, , , r I e h + h dx=—= I I "7= I e h dy\e tl dx 
This identity will be presupposed in all that follows. 
Let us now consider a particle wandering from the origin for a given 
period of time, say from t = 0 to t = 1 . Its position will then be a function 
of the time, say x=f(t). There are certain quantities — functionals — 
which may depend on the whole set of values of / from t = 0 to t = 1. If 
we take a large number of particles (i.e. a large number of values of /) at 
random, it is natural to suppose that the average value of the functional 
will often approach a limit, which we may call the average value of the 
functional over its entire range. What will this average be, and how shall 
we find it? 
If F {/} is a functional depending on the values of /for only a finite num- 
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