Ergodic theory for general Markov chains

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2 total messages Started by Tobias Ryden Thu, 09 Apr 1992 14:18

#194

Author: Tobias Ryden
Date: Thu, 09 Apr 1992 14:18

32 lines
1361 bytes

For a stationary ergodic process X_n it is well known that

       N
  1   ---
  -   \
  N   /    f(X_n) -> E(f(X_1))
      ---
      n=1

with probability 1 and in L_1 provided E|f(X_1)| is finite. Now, let
Y_n be a Markov chain on a general state space E for which there
exists a stationary distribution v, but the initial distribution of
Y, i.e. the distribution of Y_1, need not necessarily equal v.
Under which conditions does

       N
  1   ---
  -   \
  N   /    f(Y_n) -> v(f)
      ---
      n=1

hold, where f is in L_1(v) and the convergence is w.p.1, in L_1 or in
probability?

All references are appreciated!

-- 
+- Tobias Ryden                          Internet: tobias@maths.lth.se
   Dept. of Mathematical Statistics      Tel:   int+46 -(0)46 104276  
   Lund Institute of Technology          Fax:   int+46 -(0)46 104623   
-- P.O. Box 118, S-221 00 Lund, Sweden

Re: Ergodic theory for general Markov chains

#197

Author: Rainer Sachs
Date: Fri, 10 Apr 1992 14:30

22 lines
1262 bytes

In article <1992Apr9.141813.21760@lth.se> tobias@maths.lth.se (Tobias Ryden) writes:
>Let Y_n be a Markov chain on a general state space E for which there
exists a stationary distribution v, but the initial distribution of
Y, i.e. the distribution of Y_1, need not necessarily equal v.
Under which conditions does

       N
  1   ---
  -   \
  N   /    f(Y_n) -> v(f)
      ---
      n=1

hold, where f is in L_1(v) and the convergence is w.p.1, in L_1 or in
>probability?

email bounced.  See, among many other places, RW Wolff, Stochastic modeling
and the theory of queues, section 3-6.  The theory of such averages
is considerably simpler than the theory of functions; for example
in a cyclic irreducible positive recurrent discrete time
Markov chain f may fail to converge but (1/n)sigma f (y sub i)
still converges.

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