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The distribution is known to be uniform (a result due to Weyl, I believe). An excellent reference for this (and much else) is Dym and McKean's book on harmonic analysis.

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.

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.

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 …

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 $N^{1/3},$ …

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 …

That source of all wisdom, Wikipedia, contains the proof of your assertion (without saying so) -- look in the "Background" section of the Schur's complement article -- in more generality. Since I assu …

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 …

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

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 \ …

This is really a follow-up on Suvrit's comment. There are plenty of formulas for products of theta functions, many of them found in this Iowa State report. (see particularly page 7). Whether any of th …

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).

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 …

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.