2 votes
Accepted

Showing that a martingale is constantly zero

Consider the Martingale $M_k:=W(Y_{k \wedge T})$, where $T$ is the first time $Y$ hits $i_0$, and $Y_0=i$. From $E[(M_k-M_0)^2]<\infty$, the Martingale is bounded in $L^2$, so it is uniformly ...
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2 votes
Accepted

Search for a non absorbing Markov chain

Suppose $p(0,1)=p(1,0)=1$ and $p(2,3)=p(3,3)=1$. This chain with 4 states has an absorbing state at state 3, but is not an absorbing chain because there is no way to get from state 1 to state 3. ...
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1 vote

Some question about probability (i guess using Markov chains)

To simplify notation, let me call the initial probabilities $p' \; and \;q'$ for boy and father, $p'+q' <1$, and ignore draws (which don't matter) P(boy wins a game) $=p=\frac{p'}{p'+q'}$ P(father ...
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1 vote

Which State Will the Markov Chain Go To Next?

Define these events for convenience: $A = (J_n = h)$ $B = (S_{n+1} > d)$ $C_j = (J_{n+1} = j)$ Conditioned on $J_n = h$ and $S_{n+1} > d$ (i.e. already waited $d$ time in $h$), the ...
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