> I'm sure that @graycat will
scoff at those
recommendations, but his
reading list would be
considered excessively hardcore
and time consuming
even for a graduate student
in math, which I'm
assuming you're not.
Probability and stochastic
processes based on measure
theory are not very popular
in the US, even in graduate
math departments.
Uh, scoff, scoff. Okay?
The full measure theoretic details
of stochastic processes in
continuous time can be a bit
of a challenge. That topic can
be important, e.g., for Brownian
motion and stochastic differential
equations used in mathematical
finance. Of course, there is
Karatzas and Shreve,
Brownian Motion and Stochastic Calculus
and Chung and Williams,
Introduction to Stochastic Integration.
And there's much more, especially from
Russia and France.
But, otherwise, usually in practice,
what people are interested in is either
(1) second order stationary stochastic
processes, e.g., as in electronic or
acoustical signals and noise. There
are commonly interested in power spectral
estimation, digital filtering, maybe
Wiener filtering, the fast Fourier transform,
etc. or (2) what is in, say, Cinlar,
Introduction to Stochastic Processes.
In Cinlar,
for the continuous time case,
get a good introduction to
the Poisson process (the vanilla
arrival process, e.g.,
like clicks at a Geiger counter,
new sessions
at a Web site, and much more).
Also get
what else people are mostly interested
in in practice, Markov processes
in discrete time with a
discrete state space (that is, the
values are discrete).
The case of Markov processes in
continuous time and discrete state
space is not so tough if the
jumps are driven by just a Poisson
process. But there is still more
in Cinlar.
And there are other good texts on
stochastic processes.
For (1), look at some of the
texts used by EEs. The measure
theory approach is in
Doob, Stochastic Processes,
Loeve, Probability Theory,
and several more texts by quite
good authors.
E.g., without measure theory,
can just dive in via
Blackman and Tukey, The Measurement
of Power Spectra ....
With all these sources,
are able to get by without
measure theory.
Yes, without measure theory,
at some places will have to
not ask to understand too much
and just skip over some details
to get back to the applied stuff.
But for measure theory, the
Durrett text seems to get
a student to that unusually
quickly.
For more, at the MIT Web site,
there is an on-line course
in mathematical finance that
avoids measure theory.
They want to use the
Radon-Nikodym theorem
and Ito integration but
still avoid measure theory.
Uh, the Radon-Nikodym theorem
is a generalization of the
fundamental theorem of calculus.
Once see it, it's dirt simple,
but a good proof takes a bit
or follow von Neumann's proof
that knocks it all off in one
stroke (it's in
Rudin, Real and Complex Analysis).
Probability and stochastic processes based on measure theory are not very popular in the US, even in graduate math departments.
Uh, scoff, scoff. Okay?
The full measure theoretic details of stochastic processes in continuous time can be a bit of a challenge. That topic can be important, e.g., for Brownian motion and stochastic differential equations used in mathematical finance. Of course, there is
Karatzas and Shreve, Brownian Motion and Stochastic Calculus and Chung and Williams, Introduction to Stochastic Integration. And there's much more, especially from Russia and France.
But, otherwise, usually in practice, what people are interested in is either (1) second order stationary stochastic processes, e.g., as in electronic or acoustical signals and noise. There are commonly interested in power spectral estimation, digital filtering, maybe Wiener filtering, the fast Fourier transform, etc. or (2) what is in, say, Cinlar, Introduction to Stochastic Processes.
In Cinlar, for the continuous time case, get a good introduction to the Poisson process (the vanilla arrival process, e.g., like clicks at a Geiger counter, new sessions at a Web site, and much more). Also get what else people are mostly interested in in practice, Markov processes in discrete time with a discrete state space (that is, the values are discrete).
The case of Markov processes in continuous time and discrete state space is not so tough if the jumps are driven by just a Poisson process. But there is still more in Cinlar.
And there are other good texts on stochastic processes.
For (1), look at some of the texts used by EEs. The measure theory approach is in Doob, Stochastic Processes, Loeve, Probability Theory, and several more texts by quite good authors. E.g., without measure theory, can just dive in via Blackman and Tukey, The Measurement of Power Spectra ....
With all these sources, are able to get by without measure theory. Yes, without measure theory, at some places will have to not ask to understand too much and just skip over some details to get back to the applied stuff.
But for measure theory, the Durrett text seems to get a student to that unusually quickly.
For more, at the MIT Web site, there is an on-line course in mathematical finance that avoids measure theory. They want to use the Radon-Nikodym theorem and Ito integration but still avoid measure theory. Uh, the Radon-Nikodym theorem is a generalization of the fundamental theorem of calculus. Once see it, it's dirt simple, but a good proof takes a bit or follow von Neumann's proof that knocks it all off in one stroke (it's in Rudin, Real and Complex Analysis).