Weighted $p$-radial Distributions on Euclidean and Matrix $p$-balls with Applications to Large Deviations

Abstract

A probabilistic representation for a class of weighted p-radial distributions, based on mixtures of a weighted cone probability measure and a weighted uniform distribution on the Euclidean ℓ n p -ball, is derived. Large deviation principles for the empirical measure of the coordinates of random vectors on the ℓ n p -ball with distribution from this weighted measure class are discussed. The class of p-radial distributions is extended to p-balls in classical matrix spaces, both for self-adjoint and non-self-adjoint matrices. The eigenvalue distribution of a self-adjoint random matrix, chosen in the matrix p-ball according to such a distribution, is determined. Similarly, the singular value distribution is identified in the non-self-adjoint case. Again, large deviation principles for the empirical spectral measures for the eigenvalues and the singular values are presented as an application.

Large deviations for uniform projections of $p$-radial distributions on $\ell_p^n$-balls

“…As discussed earlier, choosing W n ≡ δ 0 gives P n,p,Wn ≡ C n,p and W n ≡ Exp(1) yields P n,p,Wn ≡ U n,p , in both cases it holds for W n ∼ W n that W n /n → 0 in probability and we get back [10, Theorem C]. Hence, we can see that both C n,p and U n,p share the same LDP behaviour in high dimensions, which is in line with similar observations made for other functionals (see, e.g., [1,12,14]). Moreover, the result even implies a certain universality of the rate function, since despite the expected sensitivity of LDPs to the underlying distributions, the rate function is the same for all sequences (W n ) n∈N that share the same limiting behaviour.…”