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Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$.

Is the $E(S^{2}_{n}| \tau \geq n)$ known? Further, is the distribution of $S^{2}_{n}$ conditioned on $\tau \geq n$ of the arcsine type, or is it concentrated around the origin (as common intuition suggests due to the conditioning)?

The first question relates to earlier post: Scaling of First-passage times for Random Walk on integer lattices

Of course, $E$ denotes expected value and $S^{2}_{n}$ denotes the square of the value of $S_{n}$.

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  • $\begingroup$ The previous question you linked to started at the origin and asked for the time to exit the box. This one seems different: Where do we start? And why do you consider both positive and negative integers if 0 absorbs (so we cannot go from positive to negative)? I do not see much relationship between the previous question and this question. $\endgroup$
    – Michael
    Commented May 27, 2015 at 21:52
  • $\begingroup$ The general Wald equality might be useful for problems of this type: If $J$ is a stopping time for a sum $S_k = \sum_{i=1}^k X_i$ of iid random variables $\{X_i\}_{i=1}^{\infty}$, then $E[e^{rS_J - J\gamma(r)}]=1$, where $\gamma(r) = \log(E[e^{rX}])$. $\endgroup$
    – Michael
    Commented May 27, 2015 at 21:55
  • $\begingroup$ Note $\gamma'(r) = \frac{E[Xe^{rX}]}{E[e^{rX}]}$ and $\gamma''(r) = \frac{-E[Xe^{rX}]^2}{E[e^{rX}]^2} + \frac{E[X^2e^{rX}]}{E[e^{rX}]}$, so for $r=0$ we get $\gamma(0)=0$, $\gamma'(0)=E[X]$, $\gamma''(0)=Var(X)$. $\endgroup$
    – Michael
    Commented May 27, 2015 at 21:59
  • $\begingroup$ By differentiating Wald we get $E[(S_J - J\gamma'(r))e^{rS_J-J\gamma(r)}] = 0$, which for $r=0$ gives $E[S_J] = E[J]E[X]$ (this recovers a more common version of Wald). Differentiating again gives $E[J]Var(X)=E[(S_J-JE[X])^2]$. If $E[X]=0$ then we get $E[J]Var(X)=E[S_J^2]$, which relates to second moments as you ask about. $\endgroup$
    – Michael
    Commented May 27, 2015 at 22:02

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this problem, and it's analogue for Brownian motion have been solved as solved as they can get, which may not be as solved as you want, by the same technique, which is an eigenfunction expansion of the transition matrices (resp. kernels). The eigenfunctions are sin for both. There is a paper by mark kac from the 50's, maybe Duke Journal, doing the discrete case, but it may possibly also be found in Feller vol. 1. All the ideas for the Brownian motion case are in Port & Stone, Brownian Motion and Potential theory, but probably also Karlin & Taylor vol 2. So for example,the density of $S_N$ conditioned not to have hit etc. converges to the first eigenfunction as $N \rightarrow \infty$

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  • $\begingroup$ Thank you for your comments. Could you please give a more specific reference to the part of Feller's or Karlin and Taylor's book you refer to ? Please also note that the question does not regard asymptotic behavior, but anyhow i doubt more might be calculable. Thanks again. $\endgroup$
    – John Lotos
    Commented May 29, 2015 at 1:57
  • $\begingroup$ I don't have access to my books, but Feller vol 1 is An Introduction to Probability Theory and Its Applications, Vol. 1, by William Feller and I think this exact problem is done in the markov chain section, and that that is where I picked up the reference to to Kac paper. You'll see that the whole distribution is there but it's given as an (explicit) eigenfunction expansion. $\endgroup$
    – Michael
    Commented May 29, 2015 at 6:24

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