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Log normal distribution has finite variance, so if you subtract the mean, the magic words are "Berry-Esseen theorem". If you don't subtract the mean, the sum diverges.

Is $N$ large (where "large" is probably "bigger than five", in practice)? If it is, the central limit theorem gives a good approximation to the distribution of $\sum_{i=1}^N x_i,$ assuming some weak c …

I am not going to answer the philosophical question of "what does it mean", but for the practical question, there is the Ziggurat method of Marsaglia to generate a sample from your favorite distributi …

$$e^{-{\sqrt{2/a}}}$$ is what Mathematica says...

I must be misunderstanding the question, but $<x, y>^2$ is a sum of the terms of the form $x_i x_j y_i y_j.$ The expectation of this term vanishes, unless $i=j,$ in which case it (the . expectation) i …

All you could conceivably want to know about the subject (and many things you might not) are in Mathai + Provost, Quadratic Forms in Random Variables.

There is a trick to reduce these kinds of questions to questions about independent normals, as in my ancient preprint. For large $n,$ concentration of measure will presumably give you easy estimates.

I am not at all sure I understand the question, since the OP's reference [2] has a formula for the entropy, and has asymptotics in the equidistributed case (the latter on page 7, the former on page 5, …

If the cumulative distribution function of $X$ is $G$ while the probabiity density $g=G^\prime,$ then the probability density of $\min(X, X^\prime)$ is $2 g G.$ Similarly, if the CDF of $Y$ is $H,$ wi …

See http://arxiv.org/pdf/cond-mat/0406116v2 for a more general version of the question (the 1-dim case is considered at length).

It is a theorem of Renyi and Soulanke that the cardinality of the boundary of a convex hull of a uniformly distributed random point set of cardinality $N$ in a smooth convex set grows like $N^{1/3},$ …

This not really appropriate for MO, but the proof follows immediately from additivity of expectation. The expected length squared of the vector is $L,$ that means that the expected square of a coordin …

Since you have the same concentration properties when $d=0,$ and $A$ is a point (by rotation invariance), the answer is Yes, there is always concentration.

More of a comment to indicate how hopeless this is: Consider the edge case of $\sigma_{ij} = \lambda_i \delta_{ij}.$ The unitary matrix then is the identity matrix, so is quite far from Haar distribut …

Assuming the normal is centered with variance $s$ and the Cauchy distribution has parameters $a, b$, combining this Wikipedia page and Mathematica gives $$ \text{ConditionalExpression}\left[\frac{i \ …