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 Jan1 comment Expectation of hitting time for simple symmetric random walk to add, $S_n$ is a Markov Chain (symmetric random walk) on the integers. It is null recurrent. The proof you just gave verifies it cannot be positively recurrent. Jan1 comment Bound on sum random variables and Martingales for your question, you might want to consider something like $\exp(\sigma S_n - n\phi(\sigma))$ where $\phi(\sigma)$ is the moment generating function of $X_i$. The reason being your question has something to do with $\exp$, considering this type of martingale seems appropriate. Jan1 comment Bound on sum random variables and Martingales please considering accepting answers for other questions you asked Dec26 awarded Nice Question Dec10 comment A sequence converges if and only if every subsequence converges? It seems like no effort has been made by the OP to try to answer the question. Dec10 reviewed Approve Evaluating $\int \frac{1}{\sin x} dx$ Dec10 reviewed Leave Open A problem of a collection of subsets maximal with respect to the finite intersection property Dec10 reviewed Close A probability problem with maximum and summation Dec10 reviewed No Action Needed Upper bound for the modulus of bounded holomorphic function in the unit disk with $f(a)=0$ Dec10 reviewed Approve Dense packing of circles in unit circle, matlab code Dec10 reviewed Approve Can you help with the Method Of Eigenfunction Expansion of a Non-Homogeneous PDE problem? Dec9 awarded Caucus Nov22 reviewed No Action Needed A die whose score cannot be as before (Markov chains) Nov22 reviewed Close How $\sqrt{x}$ is an function? Nov22 asked Notational question on vector field/differential operator Nov5 comment Limit of: $\lim_{x\rightarrow\infty}\left(\dfrac{x+2}{x+1}\right)^{x/2}$ Jumped way too many steps with the last equality, even with the comment. Given that the OP asked this question, I am pretty convinced he could not have filled all these gaps himself. Nov5 comment Limit of: $\lim_{x\rightarrow\infty}\left(\dfrac{x+2}{x+1}\right)^{x/2}$ still not convinced, you are saying $\lim a^b = (\lim a)^{(\lim b)}$. Your supposedly justification does not do the job. Nov4 comment Limit of: $\lim_{x\rightarrow\infty}\left(\dfrac{x+2}{x+1}\right)^{x/2}$ Not convinced by the last equality. Nov2 reviewed Approve complicated projectile motion about throwing thing Oct29 accepted Binomial/Poisson distribution question