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

E. Gluskin; S. Kwapień

Displaying similar documents to “Tail and moment estimates for sums of independent random variables with logarithmically concave tails”

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...

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....

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 ${X_{n}, n \geq 1}$ .

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 ( $\vec{a}$ ,...

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

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,...

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.

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....

Volumetric invariants and operators on random families of Banach spaces

Piotr Mankiewicz, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

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The geometry of random projections of centrally symmetric convex bodies in $ℝ^{N}$ is studied. It is shown that if for such a body K the Euclidean ball $B ₂^{N}$ is the ellipsoid of minimal volume containing it and a random n-dimensional projection $B = P_{H} (K)$ is “far” from $P_{H} (B ₂^{N})$ then the (random) body B is as “rigid” as its “distance” to $P_{H} (B ₂^{N})$ permits. The result holds for the full range of dimensions 1 ≤ n ≤ λN, for arbitrary λ ∈ (0,1).

Positivity of integrated random walks

Vladislav Vysotsky (2014)

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

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Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n} : = \sum_{i = 1}^{n} S_{i}$ . Suppose $S_{1}$ is in the domain of normal attraction of an $α$ -stable law with $1 l t; α \leq 2$ . Assuming that $S_{1}$ is either right-exponential (i.e. $ℙ (S_{1} g t; x | S_{1} g t; 0) = e^{- a x}$ for some $a g t; 0$ and all $x g t; 0$ ) or right-continuous (skip free), we prove that $ℙ {A_{1} g t; 0, \dots, A_{N} g t; 0} \sim C_{α} N^{1 / (2 α) - 1 / 2}$ as $N \to \infty$ , where $C_{α} g t; 0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

The absolute continuity of the invariant measure of random iterated function systems with overlaps

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

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We consider iterated function systems on the interval with random perturbation. Let $Y_{ε}$ be uniformly distributed in [1-ε,1+ ε] and let $f_{i} \in C^{1 + α}$ be contractions with fixpoints $a_{i}$ . We consider the iterated function system $Y_{ε} f_{i} + a_{i} (1 - Y_{ε}) ⁿ_{i = 1}$ , where each of the maps is chosen with probability $p_{i}$ . It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection...

On uniqueness of distribution of a random variable whose independent copies span a subspace in $L_{p}$

S. Astashkin, F. Sukochev, D. Zanin (2015)

Studia Mathematica

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Let 1 ≤ p < 2 and let $L_{p} = L_{p} [0, 1]$ be the classical $L_{p}$ -space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable $f \in L_{p}$ spans in $L_{p}$ a subspace isomorphic to some Orlicz sequence space $l_{M}$ . We give precise connections between M and f and establish conditions under which the distribution of a random variable $f \in L_{p}$ whose independent copies span $l_{M}$ in $L_{p}$ is essentially unique.

A Note on the Men'shov-Rademacher Inequality

Witold Bednorz (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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We improve the constants in the Men’shov-Rademacher inequality by showing that for n ≥ 64, $E (s u p_{1 \leq k \leq n} | \sum_{i = 1}^{k} X_{i} | ² \leq 0.11 (6.20 + l o g ₂ n) ²$ for all orthogonal random variables X₁,..., Xₙ such that $\sum_{k = 1}^{n} E | X_{k} | ² = 1$ .

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...

On the singular values of random matrices

Shahar Mendelson, Grigoris Paouris (2014)

Journal of the European Mathematical Society

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We present an approach that allows one to bound the largest and smallest singular values of an $N \times n$ random matrix with iid rows, distributed according to a measure on $ℝ^{n}$ that is supported in a relatively small ball and linear functionals are uniformly bounded in $L_{p}$ for some $p > 8$ , in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of $1 \pm c \sqrt{n / N}$ not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product...

On the Maximal Lévy-Ottaviani Inequality for Sums of Independent and Dependent Random Vectors

Zbigniew S. Szewczak (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that the sums $S_{k}$ of independent random vectors satisfy $P (m a x_{1 \leq k \leq n} ∥ S_{k} ∥ > 3 t) \leq 2 m a x_{1 \leq k \leq n} P (∥ S_{k} ∥ > t)$ , t ≥ 0.

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

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We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition $(T^{'})$ . We show that for every $ϵ > 0$ and $n$ large enough, the annealed probability of linear slowdown is bounded from above by $exp (- {(log n)}^{d - ϵ})$ . This bound almost matches the known lower bound of $exp (- C {(log n)}^{d})$ , and signiﬁcantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched...

On the geometry of proportional quotients of $l ₁^{m}$

Piotr Mankiewicz, Stanisław J. Szarek (2003)

Studia Mathematica

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We compare various constructions of random proportional quotients of $l ₁^{m}$ (i.e., with the dimension of the quotient roughly equal to a fixed proportion of m as m → ∞) and show that several of those constructions are equivalent. As a consequence of our approach we conclude that the most natural “geometric” models possess a number of asymptotically extremal properties, some of which were hitherto not known for any model.

Random walks on co-compact fuchsian groups

Sébastien Gouëzel, Steven P. Lalley (2013)

Annales scientifiques de l'École Normale Supérieure

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It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$ . It is also shown that Ancona’s inequalities extend to $R$ , and therefore that the Martin boundary for $R$ -potentials coincides with the natural geometric boundary $S^{1}$ , and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic...

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

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We consider random walk on a discrete torus $E$ of side-length $N$ , in sufficiently high dimension $d$ . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u N^{d}$ . We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $log N$ . Moreover, this connected component occupies a...

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

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We study a continuous time growth process on the $d$ -dimensional hypercubic lattice $𝒵^{d}$ , which admits a phenomenological interpretation as the combustion reaction $A + B \to 2 A$ , where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

The spread of a catalytic branching random walk

Philippe Carmona, Yueyun Hu (2014)

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

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We consider a catalytic branching random walk on $ℤ$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_{n}$ : For some constant $α$ , $\frac{M_{n}}{n} \to α$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_{n} - α n$ as $n$ goes to infinity.

Persistence of iterated partial sums

Amir Dembo, Jian Ding, Fuchang Gao (2013)

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

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Let $S_{n}^{(2)}$ denote the iterated partial sums. That is, $S_{n}^{(2)} = S_{1} + S_{2} + \dots + S_{n}$ , where $S_{i} = X_{1} + X_{2} + \dots + X_{i}$ . Assuming $X_{1}, X_{2}, ..., X_{n}$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $p_{n}^{(2)} : = ℙ (max_{1 \leq i \leq n} S_{i}^{(2)} l t; 0) \leq c \sqrt{\frac{𝔼 | S_{n + 1} |}{(n + 1) 𝔼 | X_{1} |}},$ with $c \leq 6 \sqrt{30}$ (and $c = 2$ whenever $X_{1}$ is symmetric). The converse inequality holds whenever the non-zero $min (- X_{1}, 0)$ is bounded or when it has only finite third moment and in addition $X_{1}$ is squared integrable. Furthermore, $p_{n}^{(2)} ≍ n^{- 1 / 4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_{i}$ . In contrast, we show that for any $0 l t; γ l t; 1 / 4$ there exist integrable,...

Asymptotic rate of convergence in the degenerate U-statistics of second order

Olga Yanushkevichiene (2010)

Banach Center Publications

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Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T (X ₁, . . ., X ₙ) = n^{- 1} \sum_{1 \leq i L e t q_{i} (i \geq 1) b e e i g e n v a l u e s o f t h e H i l b e r t - S c h m i d t o p e r a t o r a s s o c i a t e d w i t h t h e k e r n e l h (x, y), a n d q ₁ b e t h e l a r g e s t i n a b s o l u t e v a l u e o n e . W e p r o v e t h a t}$ Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12) $,$ where $G_{i}$ , 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β^{'} : = E | h (X, X ₁) | ³ + {E | h (X, X ₁) |}^{18 / 5} + E | g (X) | ³ + {E | g (X) |}^{18 / 5} + 1 < \infty$ .