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Arkadiusz gives the answer in the case of two independent Gaussians. A simple technique to reduce the correlated case to the uncorrelated is to diagonalize the system. The intuition which I use is t …

If you leave the realm of abstract probability spaces and focus on probability in Banach spaces, there's a lot of geometry to take advantage of. Here's an example. Let $X$ be a Banach space, and let …

Your double subscripts are extraneous. Let's consider a simpler situation, where we have a single family of random variables $\{X_i\}$. As Yuri Bakhtin says above, your condition is not sufficient f …

As Leandro suggested in the comments, this should follow a power-law decay in $n$. However, Hara and Slade's rigorous work using lace expansions is only valid for dimensions $\ge 19$. Much of the ri …

Edit (Jan 28): As Didier points out in the comments, I made a mistake in my application of Chebyshev's inequality. Didier and fedja already have gave you some great answers, but I'd like to go a l …

That was extremely difficult to parse. Until LaTeX support isn't yet enabled, please try to more simply! (e.g. don't use \left and \right) Consider the independent case with t constant. I was hopi …

Here's my rewording of your question. Think of Y below as log|X|. "Let φ(t) = EetY be the moment-generating function of Y. Suppose that for any C > 0, φ(t) ≤ ebt + Ct². If Yi are identical cop …

Let $X$ and $Y$ be two positive random variables with $Y < X$; these may be highly correlated. I would like a reasonable condition on $X$ and $Y$ so that the ratio $X/Y$ has a finite moment-generatin …

Let $X_i$ be a sequence of identically-distributed random variables with finite-range dependence (i.e. there exists $I$ such that if $|i-i'| \ge I$, then $X_i$ and $X_{i'}$ are independent), and a fin …

I think Lyons answers this in Russell Lyons - Strong Laws of Large Numbers for Weakly Correlated Random Variables.

There is nothing mathematically wrong with your notation. However, I don't like it, because $Z'$ suggests that you are taking a derivative with respect to the background randomness. I would rather w …

Let $X_n$ be a sequence of iid positive random variables. Assume that $X_n$ has finite $\alpha$th moment for some value $\alpha \in (0,1)$, but infinite first moment. Assume also that the reciprocal $ …

Here's another simple example, in a similar vein as has2's above. Let $X_n$ be a sequence of independent, identically-distributed exponential variables, i.e., $$\mathbb P(X_n > u) = e^{-\lambda u},$$ …

Here's a concrete example of an atomless measure. Let $f \in L^1$ be an integrable function with total mass 1 (i.e. $\int_0^1 f = 1$). Define $$\mathbb P(A) = \int_A f(x) ~dx$$ for any Borel set $A$ …

The diameter is defined as $$d(G') = \sup_{x,y \in G} \inf_{\gamma} \sum_{e \in \gamma} w_{e},$$ where the infimum is over all paths $\gamma$ connecting $x$ to $y$, and the sum is over the edges $e$ w …