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Unformatted text preview: e. The deﬁnition traces its roots to the notion
of superharmonic functions whose values at a point exceed the average value
on balls centered around the point. If we reverse the sign and suppose
E (Mn+1 Mn |Av ) 0 then Mn is called a submartingale with respect to Xn . A simple modiﬁcation of the proof for Example 5.2 shows that if µ 0, then Sn deﬁnes a
supermartingale, while if µ 0, then Sn is a submartingale.
The next result will lead to a number of examples.
Theorem 5.5. Let Xn be a Markov chain with transition probability p and let
f (x, n) be a function of the state x and the time n so that
f (x, n) = X p(x, y )f (y, n + 1) y Then Mn = f (Xn , n) is a martingale with respect to Xn . In particular if
h(x) = y p(x, y )h(y ) then h(Xn ) is a martingale.
Proof. By the Markov property and our assumption on f
E (f (Xn+1 , n + 1)|Av ) = X p(xn , y )f (y, n + 1) = f (xn , n) y which proves the desired result.
The next two examples begin to explain our interest in Theorem 5.5.
Example 5.3. Gambler’s ruin. Let X1 , X2 , . . . be independent with
P (Xi =...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
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