Proof for Markov Chain steady-state probabilities

neeraj2608 · Joined: 05 Aug 2007 Posts: 1

Hi,
I was wondering whether there exists any rigorous proof for the
following two premises of Markov chain theory:
a. For a regular transition matrix T, T^n -> p as n -> infinity where
p is a matrix of the form [v, v, ..., v] with v being a constant
vector
b. For any state vector X, (T^n * X) -> q where q is a fixed
probability vector (its entries add up to 1), all of whose entries are
positive.
From whatever material I've found, these two premises seem to be based
on observation/induction and not proof.

Merci,
Neeraj.

Note: This post has been made to the following groups: sci.math,
uk.education.maths, sci.math.research, sci.stat.math, sci.physics

Archived from group: sci>math

Stephen J. Herschkorn · Joined: 05 Aug 2007 Posts: 3

neeraj2608@gmail.com wrote:

>I was wondering whether there exists any rigorous proof for the
>following two premises of Markov chain theory:
>a. For a regular transition matrix T, T^n -> p as n -> infinity where
>p is a matrix of the form [v, v, ..., v] with v being a constant
>vector
>b. For any state vector X, (T^n * X) -> q where q is a fixed
>probability vector (its entries add up to 1), all of whose entries are
>positive.
>From whatever material I've found, these two premises seem to be based
>on observation/induction and not proof.
>
>

You must not have been looking in the right places then. See any
standard rigorous text on stochastic processes, e.g., Karlin and
Taylor, Ross (my favorite), or Wolff.

>Merci,
>Neeraj.
>
>Note: This post has been made to the following groups: sci.math,
>uk.education.maths, sci.math.research, sci.stat.math, sci.physics
>
>
My reader allows me to post to at most four newsgroups at a time. I
have removed sci.math.research and sci.physics and added
alt.sci.math.probability

--
Stephen J. Herschkorn sjherschko@netscape.net

Gerry Myerson · Joined: 05 Aug 2007 Posts: 1

In article ,
"Stephen J. Herschkorn" wrote:

> neeraj2608@gmail.com wrote:
>
> >I was wondering whether there exists any rigorous proof for the
> >following two premises of Markov chain theory:
> >a. For a regular transition matrix T, T^n -> p as n -> infinity where
> >p is a matrix of the form [v, v, ..., v] with v being a constant
> >vector
> >b. For any state vector X, (T^n * X) -> q where q is a fixed
> >probability vector (its entries add up to 1), all of whose entries are
> >positive.
> >From whatever material I've found, these two premises seem to be based
> >on observation/induction and not proof.
> >
> >
>
> You must not have been looking in the right places then. See any
> standard rigorous text on stochastic processes, e.g., Karlin and
> Taylor, Ross (my favorite), or Wolff.

This is one of many places where you can state and use a result
at one level of mathematical sophistication but the proof lies at
a considerably higher level. Therefore it is common for people to
teach the concept/algorithm/whatever without giving the proof.
I do it all the time. But I generally stress that there is a proof so
the students don't get the impression that the math is based
solely on observation.

I've had to trim the newsgroups to those supported by
my server.

--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)

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