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

Zbigniew S. Szewczak

Displaying similar documents to “On the Maximal Lévy-Ottaviani Inequality for Sums of Independent and Dependent Random Vectors”

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

Andrzej Kłopotowski

Similarity:

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

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

Similarity:

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

Estimates of $L_{p}$ norms for sums of positive functions

Ilgiz Kayumov (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

We present new inequalities of $L_{p}$ norms for sums of positive functions. These inequalities are useful for investigation of convergence of simple partial fractions in $L_{p} (ℝ)$ .

Geometrically strictly semistable laws as the limit laws

Marek T. Malinowski (2007)

Discussiones Mathematicae Probability and Statistics

Similarity:

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....

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

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

Studia Mathematica

Similarity:

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.

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

Similarity:

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

Positivity of integrated random walks

Vladislav Vysotsky (2014)

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

Similarity:

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.

A Note on the Men'shov-Rademacher Inequality

Witold Bednorz (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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

The spread of a catalytic branching random walk

Philippe Carmona, Yueyun Hu (2014)

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

Similarity:

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.

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

Piotr Mankiewicz, Stanisław J. Szarek (2003)

Studia Mathematica

Similarity:

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 Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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....

Persistence of iterated partial sums

Amir Dembo, Jian Ding, Fuchang Gao (2013)

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

Similarity:

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,...

On the length of rational continued fractions over $_{q} (X)$

S. Driss (2015)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

Let $_{q}$ be a finite field and $A (Y) \in_{q} (X, Y)$ . The aim of this paper is to prove that the length of the continued fraction expansion of $A (P); P \in_{q} [X]$ , is bounded.

Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

Albert Baernstein II, Robert C. Culverhouse (2002)

Studia Mathematica

Similarity:

Let $X = \sum_{i = 1}^{k} a_{i} U_{i}$ , $Y = \sum_{i = 1}^{k} b_{i} U_{i}$ , where the $U_{i}$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_{i}, b_{i}$ are real constants. We prove that if $b ²_{i}$ is majorized by $a ²_{i}$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L ² - L^{p}$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

On the $k$ -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

Similarity:

For any positive integer $k \geq 3$ , it is easy to prove that the $k$ -polygonal numbers are $a_{n} (k) = (2 n + n (n - 1) (k - 2)) / 2$ . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$ -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S (a_{n} (k) {\bar{a}}_{m} (k), p)$ for $k$ -polygonal numbers with $1 \leq m, n \leq p - 1$ , and give an interesting computational formula for it.

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

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

Similarity:

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...

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

Similarity:

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

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

Similarity:

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...

Asymptotic behavior of a stochastic combustion growth process

Alejandro Ramírez, Vladas Sidoravicius (2004)

Journal of the European Mathematical Society

Similarity:

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...

On bilinear forms based on the resolvent of large random matrices

Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet (2013)

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

Similarity:

Consider a $N \times n$ non-centered matrix $𝛴_{n}$ with a separable variance profile: $𝛴_{n} = \frac{D_{n}^{1 / 2} X_{n} {\tilde{D}}_{n}^{1 / 2}}{\sqrt{n}} + A_{n} .$ Matrices $D_{n}$ and ${\tilde{D}}_{n}$ are non-negative deterministic diagonal, while matrix $A_{n}$ is deterministic, and $X_{n}$ is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by $Q_{n} (z)$ the resolvent associated to $𝛴_{n} 𝛴_{n}^{*}$ , i.e. $Q_{n} (z) = {(𝛴_{n} 𝛴_{n}^{*} - z I_{N})}^{- 1} .$ Given two sequences of deterministic vectors $(u_{n})$ and $(v_{n})$ with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:...

On families of weakly dependent random variables

Tomasz Łuczak (2011)

Banach Center Publications

Similarity:

Let $ₙ^{(k)}$ be a family of random independent k-element subsets of [n] = 1,2,...,n and let $(ₙ^{(k)}, ℓ) = ₙ^{(k)} (ℓ)$ denote a family of ℓ-element subsets of [n] such that the event that S belongs to $ₙ^{(k)} (ℓ)$ depends only on the edges of $ₙ^{(k)}$ contained in S. Then, the edges of $ₙ^{(k)} (ℓ)$ are ’weakly dependent’, say, the events that two given subsets S and T are in $ₙ^{(k)} (ℓ)$ are independent for vast majority of pairs S and T. In the paper we present some results on the structure of weakly dependent families of subsets obtained in this way. We...

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

Similarity:

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,...

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

Similarity:

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...

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

Similarity:

We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ as a runs through $F^{\times}$ , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $| F^{\times} |$ ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...