From Osher Doctorow
"Stochastic" just means "probability" and "process" in mathematical
probability-statistics is intuitively something that happens at 2 or
more times and is usually labelled by time subscripts which can be for
example integers such as X1, X2, X3, ... for random variables Xi, i =
1, 2, 3, ..., or even arbitrary time subscripts in the real (often
nonnegative) numbers {X_t, t in an index set I} or {Xt: t in an index
set I}. It is often assumed that these random variables are in some
sense the "same but changing object", so that they typically but not
always have the same mean or the same variance or some other same
characteristic. Readers may have noticed that I typically drop time
subscripts (for one thing, they're rather ponderous to work with),
which is nonconformist for mathematical probability-statistics.
So in a sense mathematicians and mathematical physicists and
theoretical physicists (the second call themselves mathematicians, the
third physicists usually) in the mainstream have already discovered
that certain properties of random variables change in time or could
theoretically change in time, including means, variances, correlations,
covariance functions, etc. However, for a physicist especially to use
this fact is a bit dangerous for a rather curious reason:
mathematicians have a tendency to over-abbreviate syntax much more than
semantics.
What am I saying? Well, I'm saying that mathematicians tend to have
"formula-shortening" or "formula-simplifying" fads without much
simplification in the underlying meanings or fundamental ideas.
Another way to say this is: they tend to like to simplify the algebra
more than the fundamental meanings behind the algebra.
Take for example statistical independence, a mainstream concept that
goes back to a long, long time ago. Mathematicians largely got
interested in it because it simplifies formulas period, especially in n
(nonnegative integer arbitrary) dimensions. For example, the
multivariate probability density function:
1) f(x1, x2, ..., xn)
for n random variables X1 to Xn simplifies under the independence
assumption to:
2) fX1(x1)fX2(x2)...fXn(xn)
and if we add the identically distributed assumption which means that
the Xi all have the same marginal density fX1 = fX2 = ... = fXn, then
(2) simplifies for x1 = x2 = ... = xn to:
3) [fX1(x1)]^n
Before physicists laugh, remember how convenient Newton's equations are
because of their brevity typically, as for example F = d(mv)/dt or the
special case F = ma or F = Gm1m2/r^2.
What could possibly go wrong with all this in the "best of all possible
worlds"? Well, in our world at least some things have in fact gone
wrong. It turns out that Markov chains and their relatives Markov
processes resemble independent random variables in stochastic processes
in that their equations simplify somewhat like (2) or (3) except for
involving "given" quantities. And guess what? Mainstream mathematical
probability-statistics especially in interdisciplinary applications has
largely been built out of Markov chains and Markov processes. Because
they look simple in equations or "formulas". I kid you not (I'm not
joking).
Take a look under "Non-Markov processes" and related keywords in Front
for the Mathematics arXiv.
Osher Doctorow
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