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Assuming $Y_{2i+1} + Y_{2i+2}$ is independent of $Y_{2i+3} + Y_{2i + 4}$, then the answer to this question answers yours in the affirmative. To prevent non-degeneracy, your condition that "distributio …

$X/|X|$ is almost surely a uniformly random element on the unit sphere of dimension $n-1$; this is not the same thing as a rotation, which is a matrix. Since a multivariate standard Gaussian vector is …

Since your covariance matrix is merely a multiple of the identity matrix, the result for standard normal should be easily adapted to your case by a scaling argument. I suspect this is a homework probl …

Consider a circle of length $t+\ell$. Then I think your problem is asking if I drop $N$ points uniformly at random onto that circle, what is the probability that at least $m$ of them are in an interva …

Say I have $X_{ij}$, $j \le i$ with the property that $X_{ij}$ are centered and identically distributed and $E(X_{ij} X_{ij'}) = o(\exp(-i)))$. Then does $\sum_j X_{ij}$ have Gaussian domain of attrac …

Given $n$ independent uniformly distributed points on $S^2$, what's the distribution of the distance between two closest points? Consider $n$ iid uniform points on $S^1$, $Y_1, \ldots, Y_n$, in count …

I am looking for a finitely parameterized family of non-atomic distributions $D(\vec{\lambda})$, $\vec{\lambda} \in \mathbb{R}^k$ for some finite $k$, such that if $X \sim D(\vec{\lambda}_1)$ and $Y \ …

Besides some trivial lower bound because of the range constraint, there is not much you can say here. I can let $Y$ be $X$ with probability $> k$ for each $X$ value and anything else otherwise. There …

Let $\Delta_n$ be the set of all probability vectors on $n$ points, also known as the $n$-simplex. Let $x, y \in \Delta_n$ be two probability vectors, that is, $\sum_{i=1}^n x_i = 1$ and $x_i \geq 0$, …

Unless there is something I didn't understand about your Brownian motion statement, the result is not true in general. Consider even a deterministic example, where $X_t^n = \frac{1}{2n} 1_{|t| < n}$. …

Since entropy distance between two probability measures bounds total variation distance, and since total variation distance is basically the $L^1$ distance of density functions, my guess is that entro …

The answer is given in terms of inclusion exclusion principle, much as the solution for coupon collector's problem. Let $p_t$ be the probability that at time $t$ there are still two vertices with no e …

As suggested by my coauthor, and proved in the same paper linked in the question, the conjectured upper bound is indeed true. I reproduce the proof here for completeness: Let $p = TV(x, y)$. We want t …

If you don't require anything else besides exchangeable, then I guess not much can be said. Since exchangeability includes the trivial case of completely coupled random variables. Say consider the fol …

Wrong solution. See James' below. I'll just add that to show independence for say $X_{\sigma(1)},X_{\sigma(2)}$, $E(E(f(X_{\sigma(1)}) g(X_{\sigma(2)})|X_{\sigma(1)})) = E( f(X_{\sigma(1)}) E(g(X_{\si …