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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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    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...)2012-12-13

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