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I assume you meant $\frac{1}{n}\sum_{t=1}^n f(X_t)$ converges to $\mathrm{E}_\pi (f)$. I'll also quibble and point out that the (classical) CLT doesn't give error rates in terms of n, but the Berry-Es …

Any mixture of Gaussians has a density, which limits then sense in which a statement like you want to make can be true. The statement you propose doesn't make sense (in part) since a distribution is …

Assuming I'm interpreting your notation correctly, with high probability it is known that $\lambda_{\min}(W_n) \ge c/n$ for some absolute constant $c$ (see Edelman, "Eigenvalues and condition numbers …

You can do something with Talagrand's inequality for at least some normalizations. The simplest case would be if the $X_i$ are mean 0 and bounded, say $|X_i| \le 1$ almost surely, and you're looking …

Here's a quick argument to get something in the direction of what you want, but rather weaker than you asked for. First of all, using the Cauchy-Schwarz inequality, $$ \mathbb{E} d_H(P,\hat{P}_n) \le …

Your reasoning looks right, although I'm not that familiar with the exact notation you're using, except that the $v_i$ should be in the denominator, not the numerator. In the second case the answer i …

If by uniform measure you mean $(n-1)$-dimensional Hausdorff measure on the sphere, the answer is no. As a consequence of the results of this paper by Barthe, Csörnyei, and Naor, under mild regularit …

Stein's method typically gives good Berry-Esseen type bounds for smooth test functions. See Chapter III of Stein's book (entirely viewable in Google Books). For example, specializing to your case of …

While it goes back more than a decade, I think Talagrand's "generic chaining"/"majorizing measures without measures" approach to bounding suprema of stochastic processes could be considered a striking …