Given a symmetric simple random walk on the line starting at the origin such that the probability of going one unity to right and left at each step is $1/2$, I would like to compute $$\mathbb{E}(T_a^2)$$ where $T_a=\min \{n:S_n\in \{-a,a\}\}$ for integer $a>0$, but I have no clue from where to start. I know that one can compute its first moment using Wald's equality once proved it is finite. Is it possible to approach the question using this result in some way? I would appreciate some hint or reference.
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1Using $E[(T+1)^2]=E[T^2]+2E[T]+1$, you can derive an inhomogeneous recurrence relation using the total expectation formula and then solve it and apply the boundary conditions. – Ian Oct 11 '21 at 03:51
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1This is basically a duplicate of a question of mine by the way (my third question ever). I will put it in the close vote tomorrow if this hasn't either been closed or deleted or answered by then. – Ian Oct 11 '21 at 03:54
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@Ian This one – John Barber Oct 11 '21 at 03:56
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@JohnBarber That's the one. – Ian Oct 11 '21 at 03:57
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Ah, you're right! I somehow managed to misread "${-a,+a}$" as "$[-a,+a]$". – John Barber Oct 11 '21 at 04:54