Weighted p-radial distributions on Euclidean and matrix p-balls with applications to large deviations

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 for W n ∼ W n we have W n /n → 0 in probability and we recover [10,Theorem C]. Hence, we can see that 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.…”

Hölder's inequality and its reverse—A probabilistic point of view

“…The ansatz ( 9) has inspired further generalization by Kaufmann and Thäle [17] who have introduced a homogeneous factor to the joint density of (Y i ) n i =1 in order to also take into account eigenvalues or singular values of random matrices in Schatten classes. A different route has been taken by Heiny, Johnston and Prochno [13], who have considered random vectors of the form X = RΘ, where R and Θ are independent, R ∈ [0, ∞) almost surely, and Θ ∼ σ (n) 2 ; note that the distribution of X is invariant under orthogonal transformations.…”

“…Therefore we have to ensure that n V (2) n − (V (1) n ) 2 + p n≥k 2 converges in distribution, and we have to determine its limit. Now by both (18) and (17) n V (2) n − (V (1) n…”