Disjointification of martingale differences and conditionally independent random variables with some applications

Sergey Astashkin; Fedor Sukochev; Chin Pin Wong

Displaying similar documents to “Disjointification of martingale differences and conditionally independent random variables with some applications”

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

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

Similarity:

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.

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,...

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

Rafał M. Łochowski (2006)

Banach Center Publications

Similarity:

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...

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

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....

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

Nguyen Van Quang, Pham Tri Nguyen (2016)

Applications of Mathematics

Similarity:

The aim of the paper is to establish strong laws of large numbers for sequences of blockwise and pairwise $m$ -dependent random variables in a convex combination space with or without compactly uniformly integrable condition. Some of our results are even new in the case of real random variables.

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

Sharp Ratio Inequalities for a Conditionally Symmetric Martingale

Adam Osękowski (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let f be a conditionally symmetric martingale and let S(f) denote its square function. (i) For p,q > 0, we determine the best constants $C_{p, q}$ such that $s u p_{n} {(| f ₙ |}^{p} {) / (1 + S ₙ ² (f))}^{q} \leq C_{p, q}$ . Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2,... and q > 0, we determine the best constants $C_{N, q}^{'}$ such that $s u p_{n} (f ₙ^{2 N - 1}) {(1 + S ₙ ² (f))}^{q} \leq C_{N, q}^{'}$ . These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if...

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

Similarity:

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.

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

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

Similarity:

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 limiting velocity of random walks in mixing random environment

Xiaoqin Guo (2014)

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

Similarity:

We consider random walks in strong-mixing random Gibbsian environments in $ℤ^{d}$ , $d \geq 2$ . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment ( (2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ( $d \geq 5$ ).

Almost sure limit theorems for dependent random variables

Michał Seweryn (2010)

Banach Center Publications

Similarity:

For a sequence of dependent random variables ${(X_{k})}_{k \in ℕ}$ we consider a large class of summability methods defined by R. Jajte in [] as follows: For a pair of real-valued nonnegative functions g,h: ℝ⁺ → ℝ⁺ we define a sequence of “weighted averages” $1 / g (n) \sum_{k = 1}^{n} (X_{k}) / h (k)$ , where g and h satisfy some mild conditions. We investigate the almost sure behavior of such transformations. We also take a close look at the connection between the method of summation (that is the pair of functions (g,h)) and the coefficients that...

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....

Sharp moment inequalities for differentially subordinated martingales

Adam Osękowski (2010)

Studia Mathematica

Similarity:

We determine the optimal constants $C_{p, q}$ in the moment inequalities ${| | g | |}_{p} \leq C_{p, q} {| | f | |}_{q}$ , 1 ≤ p< q< ∞, where f = (fₙ), g = (gₙ) are two martingales, adapted to the same filtration, satisfying |dgₙ| ≤ |dfₙ|, n = 0,1,2,..., with probability 1. Furthermore, we establish related sharp estimates ||g||₁ ≤ supₙΦ(|fₙ|) + L(Φ), where Φ is an increasing convex function satisfying certain growth conditions and L(Φ) depends only on Φ.

Moment Inequality for the Martingale Square Function

Adam Osękowski (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Consider the sequence ${(C ₙ)}_{n \geq 1}$ of positive numbers defined by C₁ = 1 and $C_{n + 1} = 1 + C ₙ ² / 4$ , n = 1,2,.... Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound |Mₙ|≤ Cₙ Sₙ(M), n=1,2,..., and show that for each n, the constant Cₙ is the best possible.

Volumetric invariants and operators on random families of Banach spaces

Piotr Mankiewicz, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

Similarity:

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).

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

Similarity:

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.

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy (2009)

Bulletin de la Société Mathématique de France

Similarity:

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $log t$ . In the quenched setting, we also sharply estimate the distribution of the walk at time $t$ .

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

α-stable random walk has massive thorns

Alexander Bendikov, Wojciech Cygan (2015)

Colloquium Mathematicae

Similarity:

We introduce and study a class of random walks defined on the integer lattice $ℤ^{d}$ -a discrete space and time counterpart of the symmetric α-stable process in $ℝ^{d}$ . When 0 < α <2 any coordinate axis in $ℤ^{d}$ , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.

Universality for random tensors

Razvan Gurau (2014)

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

Similarity:

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 distance between ⟨X⟩ and $L^{\infty}$ in the space of continuous BMO-martingales

Litan Yan, Norihiko Kazamaki (2005)

Studia Mathematica

Similarity:

Let X = (Xₜ,ℱₜ) be a continuous BMO-martingale, that is, ${| | X | |}_{B M O} \equiv s u p_{T} | | E [| X_{\infty} - X_{T} | | ℱ_{T} {] | |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Define the critical exponent b(X) by $b (X) = b > 0 : s u p_{T} | | E [e x p (b ² (⟨ X ⟩_{\infty} - ⟨ X ⟩_{T})) | ℱ_{T}] {| |}_{\infty} < \infty$ , where the supremum is taken over all stopping times T. Consider the continuous martingale q(X) defined by $q (X) ₜ = E [⟨ X ⟩_{\infty} | ℱ ₜ] - E [⟨ X ⟩_{\infty} | ℱ ₀]$ . We use q(X) to characterize the distance between ⟨X⟩ and the class $L^{\infty}$ of all bounded martingales in the space of continuous BMO-martingales, and we show that the inequalities $1 / 4 d ₁ (q (X), L^{\infty}) \leq b (X) \leq 4 / d ₁ (q (X), L^{\infty})$ hold for every continuous BMO-martingale X. ...

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.

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

Similarity:

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. ...

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

Similarity:

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)...

A law of the iterated logarithm for general lacunary series

Charles N. Moore, Xiaojing Zhang (2012)

Colloquium Mathematicae

Similarity:

We prove a law of the iterated logarithm for sums of the form $\sum_{k = 1}^{N} a_{k} f (n_{k} x)$ where the $n_{k}$ satisfy a Hadamard gap condition. Here we assume that f is a Dini continuous function on ℝⁿ which has the property that for every cube Q of sidelength 1 with corners in the lattice ℤⁿ, f vanishes on ∂Q and has mean value zero on Q.