Strong laws of large numbers for sequences of blockwise and pairwise $m$-dependent random variables in metris spaces

Nguyen Van Quang; Pham Tri Nguyen

Displaying similar documents to “Strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in metris spaces”

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,...

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

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....

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

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).

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

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.

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.

Random fixed points for a certain class of asymptotically regular mappings

Balwant Singh Thakur, Jong Soo Jung, Daya Ram Sahu, Yeol Je Cho (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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Let (Ω, σ) be a measurable space and K a nonempty bounded closed convex separable subset of a p-uniformly convex Banach space E for p > 1. We prove a random fixed point theorem for a class of mappings T:Ω×K ∪ K satisfying the condition: For each x, y ∈ K, ω ∈ Ω and integer n ≥ 1, ⃦Tⁿ(ω,x) - Tⁿ(ω,y) ⃦ ≤ aₙ(ω)· ⃦x - y ⃦ + bₙ(ω) ⃦x -Tⁿ(ω,x) ⃦ + ⃦y - Tⁿ(ω,y) ⃦ + cₙ(ω) ⃦x - Tⁿ(ω,y) ⃦ + ⃦y - Tⁿ(ω,x) ⃦, where aₙ, bₙ, cₙ: Ω → [0, ∞) are functions satisfying certain conditions and Tⁿ(ω,x)...

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.

On the Variance Associated to a Family of Ovaloids in the Euclidean Space $E_{3}$

Giuseppe Caristi, Giovanni Molica Bisci (2007)

Bollettino dell'Unione Matematica Italiana

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In this paper we consider a random variable $h$ arising from an intersection problem between a fixed convex body $K_{0}$ and a system of random independent and uniformly distributed ovaloids in $E_{3}$ .

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...

Volume thresholds for Gaussian and spherical random polytopes and their duals

Peter Pivovarov (2007)

Studia Mathematica

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Let g be a Gaussian random vector in ℝⁿ. Let N = N(n) be a positive integer and let $K_{N}$ be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes $V_{N} : = v o l (K_{N} \cap R B ₂ ⁿ) / v o l (R B ₂ ⁿ)$ . For a large range of R = R(n), we establish a sharp threshold for N, above which $V_{N} \to 1$ as n → ∞, and below which $V_{N} \to 0$ as n → ∞. We also consider the case when $K_{N}$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both...

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...

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 $\sum_{k = 1}^{n} E | X_{k} | ² = 1$ .

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.

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.

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...

Random walks in ${(ℤ_{+})}^{2}$ with non-zero drift absorbed at the axes

Irina Kurkova, Kilian Raschel (2011)

Bulletin de la Société Mathématique de France

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Spatially homogeneous random walks in ${(ℤ_{+})}^{2}$ with non-zero jump probabilities at distance at most $1$ , with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes. ...

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 some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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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...

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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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.

Collisions of random walks

Martin T. Barlow, Yuval Peres, Perla Sousi (2012)

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

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A recurrent graph $G$ has the infinite collision property if two independent random walks on $G$ , started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in $ℤ^{d}$ with $d \geq 19$ and the uniform spanning tree in $ℤ^{2}$ all have the infinite collision property. For power-law combs and spherically...

Displaying similar documents to “Strong laws of large numbers for sequences of blockwise and pairwise m -dependent random variables in metris spaces”

Displaying similar documents to “Strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in metris spaces”