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I want to estimate exponentially the following probability:

Let $\bf{U}\in\mathbb{R}^n$ be a random vector uniformly distributed on the $n$-dimensional hypersphere, centered at the origin with radius $R$, and let $A\in\mathbb{S}_{++}^{n\times n}$ be a positive definite and deterministic matrix. I want to estimate the probability that $$ \text{Pr}(||\bf{v}-A\bf{U}||^2\approx n\alpha) = ?\ \ \ (1). $$

As example, when $A$ is an identity matrix, we have $$ \text{Pr}(||\bf{v}-\bf{U}||^2\approx n\alpha) $$ which can be interpreted as the probability that a randomly vector $\bf{U}$ on the hypersphere ($R$) will fall on the hypersphere centered around $\bf{v}$ with radius $\approx\sqrt{n\alpha}$. Equivalently, we want to find the probability that a randomly vector $\bf{U}$ on the hypersphere ($R$) would have some "correlation" coefficient (Pearson product), $\beta$, with $\bf{v}$. This probability can be calculated as the fraction between the surface area of the $n-2$ dimensional "circle" with radius $R\sin(\gamma)$ (in which $\beta = \cos(\gamma)$), and the $n$ dimensional sphere of radius $R$. It is easily can be shown that as $n\to\infty$ this probability is given by $\exp(n\ln(1-\beta^2)/2)$.

When we introduce the matrix $A$, then $\bf{Y} = AU$ lives on the the $n$-dimensional hyper-ellipsoid. Accordingly, we want to find the probability that a randomly vector $\bf{Y}$ on the hyper-ellipsoid will fall on the hypersphere centered around $\bf{v}$ with radius $\approx\sqrt{n\alpha}$. So we need to find the fraction between the surface area of the interaction of this two shapes, and the surface area of the hyper-ellipsoid.

Any suggestions how to accomplish that, or, solving the problem in different way?

Thank you!

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what is the range or distribution of v –  dexter04 Dec 5 '12 at 3:23
    
$\bf{v}\in\mathbb{R}^n$ is deterministic, and you can assume that it is a perturbed version of $\bf{AU}$. In others words, you can assume that their is an intersection between the hyper-sphere and the hyper-ellipsoid –  Josh Dec 5 '12 at 10:29
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In problems like this, it's usually convenient to approximate the random vector $\mathbf{U}$ with a Gaussian vector $\mathbf{U'}\sim {\cal N}\left(0, \frac{R}{\sqrt{n}}\cdot I_n\right)$. Then the problem reduces to a problem of estimating the density of a noncentral generalized $\chi^2$-distribution. –  Yury Dec 7 '12 at 5:53
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@Josh: I don't think there is a nice answer to your question in general. It is much easier to work with the normal distribution than with the uniform distribution on the unit sphere in particular because the former has a closed form formula and the latter doesn't. (BTW, I didn't get a notification that you replied to my message; probably because you put a space between "@" and my name.) –  Yury Dec 8 '12 at 2:56
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I'm still not sure what exactly you are looking for. Can you post all the assumptions and the desired conclusion without any ambiguities? (the sign $\approx$ alone can mean 5 or 6 different things...) –  fedja Dec 13 '12 at 5:40

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