Markov Chains and Stochastic Stability by Sean Meyn & Richard Tweedie

Author:Sean Meyn & Richard Tweedie [Meyn, Sean & Tweedie, Richard]
Language: eng
Format: epub
Published: 2011-02-19T05:00:00+00:00

=

aiEx f (Φk+i)1l(k < τB) i=0 k=0

∞

τB−1

=

aiEx f (Φk+i) .

i=0

k=0

We now have a relatively simple task in proving Theorem 11.3.11 Suppose that Φ is ψ-irreducible.

(i) If (V2) holds for a function V and a petite set C then for any B ∈ B+(X) there exists c(B) < ∞ such that

Ex[τB] ≤ V (x) + c(B),

x ∈ X.

Hence if V is bounded on A, then A is regular.

(ii) If there exists one regular set C ∈ B+(X), then C is petite and the function V = VC satisfies (V2), with V uniformly bounded on A for any regular set A.

Proof

To prove (i), suppose that (V2) holds, with V bounded on A and C

a ψa-petite set. Without loss of generality, from Proposition 5.5.6 we can assume

∞

i=0 i ai < ∞. We also use the simple but critical bound from the definition of petiteness:

1lC(x) ≤ ψa(B)−1Ka(x, B),

x ∈ X, B ∈ B+(X).

(11.27)

By Lemma 11.3.9 and the bound (11.27) we then have τB−1

Ex[τB] ≤ V (x) + bEx

1lC(Φk)

k=0

τB−1

≤ V (x) + bEx

ψa(B)−1Ka(Φk, B)

k=0

∞

τB−1

= V (x) + bψa(B)−1

aiEx

1lB(Φk+i)

i=0

k=0

∞

≤ V (x) + bψa(B)−1

(i + 1)ai

i=0

for any B ∈ B+(X), and all x ∈ X. If V is bounded on A, it follows that sup Ex[τB] < ∞,

x∈A

which shows that A is regular.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.