[R] Stochastic (transition) matrices: how to determine distributions and variance?

[Jonathan Greenberg]

(apologies for the cross-posting, and for this being a more general 
stats question rather than a specific-to-R one.  I assure you I will be 
doing the actual analysis in R :)

I am trying to determine the distribution and variance for a classic 
stochastic (transition) matrix problem such that:

let x(t) be an initial state vector consisting of counts of classes A, B 
and C:
x(t) = [A(t),B(t),C(t)]
T is the stochastic (transition) matrix for these classes consisting of 
the transition probabilities between each combination of A,B and C:

       pAA pBA pCA
T=   pAB pBB pCB
       pAC pBC pCC

By doing matrix multiplication of Tx(t) we can determine the *mean* 
counts of these classes at t+1 such that:
x mean (t+1) = Tx(t) = [A mean (t+1),B mean (t+1),C mean (t+1)]

What I want to know is what is a) what is the *distribution* of 
A(t+1),B(t+1) and C(t+1), and what is the variance around these mean 
values?  Since pXY are stochastic probabilities, it seems that the 
distribution and variance should be calculable.

Thanks!

--j

-- 

Jonathan A. Greenberg, PhD
Postdoctoral Scholar
Center for Spatial Technologies and Remote Sensing (CSTARS)
University of California, Davis
One Shields Avenue
The Barn, Room 250N
Davis, CA 95616
Cell: 415-794-5043
AIM: jgrn307, MSN: jgrn307 at hotmail.com, Gchat: jgrn307