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 Apr1 comment Finding functions from a series I disagree, for instance, if $z=-1$ then this series converges to $1$. Mar30 comment Integration over Haar Measure Dear user146290, provided that $f$ is positive or Integrable with respect to the product measure for any $X$ and $Y$ you can use the Fubini-Tonelli Theorem to compute this double integral by computing the iterated integrals, in any order you want. You could, for example, integrate over O(m) and then over O(n). Aug4 awarded Yearling Jul2 awarded Curious Jul1 revised Independence of Random Variables and Distribution Functions A mistyping correction on the first equality. Jul1 suggested approved edit on Independence of Random Variables and Distribution Functions Jun26 revised Nowhere dense set with positive Lebesgue measure edited body May14 awarded Nice Question Apr17 awarded Nice Answer Oct13 comment Show convergence in probability Hi Jack, I do not mean your hypothesis implies that $X_i$ have finite first moment, I just give you the argument in this case. Oct13 comment Recurrence for dependent random walks. Yes I decided to rewrite in order to match your answer. Oct13 comment Recurrence for dependent random walks. I changed the question. Because the first version of it was not clear and it seems that I need to be more precise about what kind of dependence I want to consider. Thanks again. Oct13 revised Recurrence for dependent random walks. Change in the dependence type. Oct13 accepted Recurrence for dependent random walks. Oct13 comment Recurrence for dependent random walks. Byron, thanks a lot for the comments and answer. You absolutely right if the steps $\{X_1,X_2,\ldots\}$ are independent. Unfortunatelly I poorly stated my problem. What I really wanted is to assume that the steps are dependent, for example, each coordinate is a Ising spin random variable interacting via an infinite range potential. In fact, I should make more precise the dependence, but I thought that this speed $cN^2$ it was enough to conclude transience independently of the type of the dependence. Oct12 answered Show convergence in probability Oct11 comment Recurrence for dependent random walks. Dear Byron, thank you for the answer. One more question: to conclude that this random walk is transient did you use some Ergodic theorem (which I guess in my case do not holds) or SLLN ( which probably holds) ? Oct11 comment Recurrence for dependent random walks. Sorry Byron, should be "some" isntead of same. Oct11 revised Recurrence for dependent random walks. edited body Oct11 answered What is an example of an open set in $\mathbb{R}^2$ which is a Cartesian product of two non-open sets in $\mathbb{R}$?