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 ...
- 13.4k
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.
...
- 13.4k
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 ...
- 29.3k
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 ...
- 15k
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