Two-parameter non-commutative Central Limit Theorem

Natasha Blitvić

Displaying similar documents to “Two-parameter non-commutative Central Limit Theorem”

Universality for random tensors

Razvan Gurau (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We prove two universality results for random tensors of arbitrary rank $D$ . We first prove that a random tensor whose entries are $N^{D}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability...

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let Ω be a countable infinite product $Ω ₁^{ℕ}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*} [X ₁] \leq l i m i n f S ₙ / n \leq l i m s u p S ₙ / n \leq E * [X ₁]$ , where $E_{*}$ and E* are the lower and upper expectations....

On the powers of Voiculescu's circular element

Ferenc Oravecz (2001)

Studia Mathematica

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The main result of the paper is that for a circular element c in a C*-probability space, $(c ⁿ, c^{n *})$ is an R-diagonal pair in the sense of Nica and Speicher for every n = 1,2,... The coefficients of the R-series are found to be the generalized Catalan numbers of parameter n-1.

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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Geometrically strictly semistable laws as the limit laws

Marek T. Malinowski (2007)

Discussiones Mathematicae Probability and Statistics

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A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_{p}$ such that $X \overset{d}{=} \sum_{k = 1}^{T (p)} X_{p, k}$ , where $X_{p, k}$ ’s are i.i.d. copies of $X_{p}$ , and random variable T(p) independent of $X_{p, 1}, X_{p, 2}, . . .$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions....

Limit theorems for sums of dependent random vectors in $R^{d}$

Andrzej Kłopotowski

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CONTENTSIntroduction.......................................................................................................................................................................... 5 I. Infinitely divisible probability measures on $R^{d}$ ....................................................................................... 6 II. The classical limit theorems for sums of independent random vectors................................................ 14 III. Convergence in law to ℒ ( $\vec{a}$ ,...

Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails

Rafał M. Łochowski (2006)

Banach Center Publications

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Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S = \sum a_{i ₁, . . ., i_{d}} X_{i ₁}^{(1)} \dots X_{i_{d}}^{(d)}$ generated by symmetric random variables $X_{i ₁}^{(1)}, . . ., X_{i_{d}}^{(d)}$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order d. Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of S and moments...

Tail and moment estimates for sums of independent random variables with logarithmically concave tails

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

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For random variables $S = \sum_{i = 1}^{\infty} α_{i} ξ_{i}$ , where $(ξ_{i})$ is a sequence of symmetric, independent, identically distributed random variables such that $l n P (| ξ_{i} | \geq t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

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Let ${X_{n, j}, 1 \leq j \leq m (n), n \geq 1}$ be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let $0 < b_{n} \to \infty$ . Conditions are given for $\sum_{j = 1}^{m (n)} X_{n, j} / b_{n} \to 0$ completely and for ${max}_{1 \leq k \leq m (n)} | \sum_{j = 1}^{k} X_{n, j} | / b_{n} \to 0$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $\sum_{j = 1}^{m (n)} X_{n, j}^{*}$ of the dependent bootstrap samples ${{X_{n, j}^{*}, 1 \leq j \leq m (n)}, n \geq 1}$ arising from a sequence of i.i.d. random variables ${X_{n}, n \geq 1}$ .

Note on the multidimensional Gebelein inequality

Marek Beśka, Agnieszka Wałachowska (2013)

Applicationes Mathematicae

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We generalize the Gebelein inequality for Gaussian random vectors in $ℝ^{d}$ .

Limit theorems for random fields

Nguyen van Thu

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CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

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We show that, given an n-dimensional normed space X, a sequence of $N = {(8 / ε)}^{2 n}$ independent random vectors ${(X_{i})}_{i = 1}^{N}$ , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ : ℝ \to ℝ^{N}$ defined by $Γ x = {(⟨ x, X_{i} ⟩)}_{i = 1}^{N}$ embeds X in $ℓ_{\infty}^{N}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{\infty}^{N}$ with asymptotically best possible relation between N, n, and ε.

Constructive quantization: approximation by empirical measures

Steffen Dereich, Michael Scheutzow, Reik Schottstedt (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article, we study the approximation of a probability measure $μ$ on $ℝ^{d}$ by its empirical measure ${\hat{μ}}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$ th moment Wasserstein metric. In the case where $2 p l t; d$ , we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada, Steffen Dereich (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the one-sided exit problem – also called one-sided barrier problem – for ( $α$ -fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $α \mapsto θ (α)$ such that the probability that any $α$ -fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{- θ (α) + o (1)}$ for large $T$ . We also investigate when the fixed level can be replaced by a different...

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

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We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$ . Restrict it to some large cube $Λ$ . Consider now $I_{Λ}$ , a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $Λ$ grows to infinity. We prove that, with probability one in the...

Probability distribution solutions of a general linear equation of infinite order, II

Tomasz Kochanek, Janusz Morawiec (2010)

Annales Polonici Mathematici

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Let (Ω,,P) be a probability space and let τ: ℝ × Ω → ℝ be a mapping strictly increasing and continuous with respect to the first variable, and -measurable with respect to the second variable. We discuss the problem of existence of probability distribution solutions of the general linear equation $F (x) = \int_{Ω} F (τ (x, ω)) P (d ω)$ . We extend our uniqueness-type theorems obtained in Ann. Polon. Math. 95 (2009), 103-114.

Coherent randomness tests and computing the $K$ -trivial sets

Laurent Bienvenu, Noam Greenberg, Antonín Kučera, André Nies, Dan Turetsky (2016)

Journal of the European Mathematical Society

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We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all $K$ -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a $K$ -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis,...

About the generating function of a left bounded integer-valued random variable

Charles Delorme, Jean-Marc Rinkel (2008)

Bulletin de la Société Mathématique de France

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We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1 - Φ (z)$ inside the unit disk, where $Φ$ is the generating function of $X$ , under some mild conditions

Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges, Sandrine Péché (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider some random band matrices with band-width $N^{μ}$ whose entries are independent random variables with distribution tail in $x^{- α}$ . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $α l t; 2 (1 + μ^{- 1})$ , the largest eigenvalues have order $N^{(1 + μ) / α}$ , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices...