From Osher Doctorow

Leo Breiman of the Rand Corporation and UCLA in his volume Probability,
Addison Wesley: Reading, Mass. 1968, has an especially good
presentation of processes with stationary independent increments, and
points out that there is a one-to-one correspondece between processes
with stationary independent increments that are continuous in
probability and characteristic functions of infinitely divisible
distributions.

A process {X(t) or X_tm t in [0, infinity)} has independent increments
if X(t + k) - X(t) or X_(t+k) - X_t is independent of the
sigma-algebra/Borel field of processes generated by the X(s) for s =
t random variables, and stationarity is like before in the previous
posts and is equivalent to the distribution of the increase not
depending on the time origin.

A characteristic function (cf) f is "infinitely divisible" iff for each
positive integer n, there is a cf f_n or fn such that:

1) f = (fn)^n, that is f(t) = (fn(t))^ for all real t

which says that f has an nth root for all positive integer n.

The characteristic function f of a random variable X is defined by:

2) f(t) = E(exp(itX)), all real t

where E(Y) for any random variable Y is the expectation or expected
value or "population mean" of Y which is the integral over the whole
real line of exp(itx) dF(x) where F is the cdf of X, and if there's a
pdf fX then dF(x) = (dF(x)/dx)dx = fX(x)dx.

The Poisson and Gaussian/normal distributions are infinitely divisible.
as is the Gamma family of distributions, but not the uniform
distribution.

Osher Doctorow

From Osher Doctorow

A process {X(t): t in I} is "continuous in probability from the right"
if X(s) converges in probability to X(t) whenever s decreases to t,
where Xn converges in probability to X if P(/Xn - X/ epsilon) -- 0
as n -- infinity for every epsilon 0. This generalizes to
continuous in probability.

Osher Doctorow