Summary: The Poisson process.
Let
I1, I2, . . . , Im, . . .
be a sequence of independent identically distributed continuous random variables. Let T0 = 0 and, for each
positive integer m, let
Tm =
m
i=0
Ii.
Evidently,
0 = T0 < T1 < · · · < Tm < · · · .
For nonnegative integers m, n with m n let
Tm,n =
m
Ii;
note that
Tn = Tm + Tm,n;
and that
Tm,n and Tm-n have the same distribution, which is to say that fTm,n = fTm-n .
Let F : R [0, 1] be such that