Thin-shell theory for rotationally invariant random simplices

Abstract

Thin-shell theory for rotationally invariant random simplices sharpening the 1/ log 1/3+o(1) n bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42(1) (2014), 146-167].

The volume of random simplices from elliptical distributions in high dimension

“…and noting that both summands on the right-hand side are independent, we conclude the following corollary from Theorem 2.1, part (2) of Remark 2.2 and (2.10). We note that this result generalizes Theorems E-F in [13] for p/n → γ, where only the case A = I n was analyzed.…”

“…The representation (1.2) immediately motivates a probabilistic analysis of random determinants of the form det(XX ⊤ ), which are in the focus of the present paper as well. We remark that the study of random determinants has a long history starting with the works in [9,10], which have later been extended by many authors, see [4,6,7,13,21,31,32], for example, as well as the references cited therein. Our main results can be summarized as follows:…”

“…Assume there exists a sequence of positive constants(σ n ) n≥1 such that lim {|X nk |<σn} ] − (E[X nk 1 {|X nk |<σn} ]) 2 = 1 , and lim n→∞ kn k=1 P(|X nk | ≥ εσ n ) = 0 , ε > 0 .Let (b n ) n≥1 be a sequence satisfyingb n = kn k=1 E[X nk 1 {|X nk |<σn} ] + o(σ n ) , n → ∞ .Then we havekn k=1 X nk − b n σ n d −→ Z ∼ N (0, 1) , n → ∞ .Theorem B.2 [26]. or[13, Theorem B.2]. Let (X nk ) n≥1,1≤k≤kn be a triangular array of realvalued random variables that are independent within rows.…”

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.…”