Prove Bernoulli Process is Markov Chain

**Jimmy_W** · September 5th 2009, 04:42 PM

Let $Z_1, Z_2, ...$ be a Bernoulli Process with success parameter p. Prove if

$Y_0, Y_1,...$ , where $Y_{k+1} = max(0, Y_k + 2Z_{k+1} - 1)$ for $k = 0,1,...$ and $Y_0 = 0$

and

$X_0, X_1,..$ where $X_{k+1} = Z_{k+1} + Z_k$ for $k = 0,1,...,$ and $X_0 = 0$

are Markov chains.

**Jimmy_W** · September 8th 2009, 06:40 PM

Does anyone have any idea how to go about these?

**pedrosorio** · September 9th 2009, 01:58 AM

$X_2 = Z_2 + Z_1$

Let us suppose $X_2 = 1$ then, either $Z_2=1 \mbox{ and } Z_1=0 \mbox{or} Z_1=0 \mbox{ and } Z_2=1$ with 0.5 probability of each case.

Given this, and knowing that $X_3 = Z_3 + Z_2$

$P(X_3 = 1|X_2 = 1) = p*0.5 + (1-p)*0.5 = 0.5$

However, if we know that $X_1 = Z_1 + Z_0 = 2$ we know that $Z_1 = 1$

Taking this into account, we get that, $X_1 = 2$ and $X_2 = 1$ , imply $Z_2 = 0$ .

Therefore,

$P(X_3 = 1 | X_2 = 1, X_1 = 2) = p$

It doesn't seem to be a Markov Chain

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