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March 29th, 2010, 12:54 PM   #1
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Prove C-K equation using markov chain definition

Could somebody tell me how to prove C-K equation in formal methods using markov chain definition?

In almost all the books that I have read, the C-K equation is said to be proved as following steps:

P{Xn = xn|X0 = x0} =
?n i=0 P{Xn = xn|Xn-1 = xn-1,Xi = xi}* P{Xn-1 = xn-1|Xi = xi} =
?n i=0 P{Xn = xn|Xn-1 = xn-1}* P{Xn-1 = xn-1|Xi = xi}

It is said that
P{Xn = xn|Xn-1 = xn-1,Xi = xi} =
P{Xn = xn|Xn-1 = xn-1}* P{Xn-1 (1)
is because we have the markov chain definition.

However, we know, markov chain is defined as:
P{Xn = xn|Xn-1 = xn-1} =
P{Xn = xn|Xn-1 = xn-1, Xn-2 = xn-2, ..., X0 = x0}

In formal method, I express Xn-1 = xn-1, Xn-2 = xn-2, ..., X0 = x0 logically as their intersection as it defined in math.

I'm not sure if the definition is correct or there are some further steps before getting (1).

Could anybody help me to answer this question? Or give me some clues (like reference, or guide)?

Thanks!
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