Displaying similar documents to “The absolute continuity of the invariant measure of random iterated function systems with overlaps”

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 : = 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 2 . Assuming that S 1 is either right-exponential (i.e. ( 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 { A 1 g t ; 0 , , A N g t ; 0 } C α N 1 / ( 2 α ) - 1 / 2 as N , 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.

Random ε-nets and embeddings in 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 : N defined by Γ x = ( x , X i ) i = 1 N embeds X in N with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into N with asymptotically best possible relation between N, n, and ε.

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 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...

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

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Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for M = S m , the m-dimensional sphere. Let Y ( B ) ; B ( S m ) be the Gaussian random measure on S m , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on S m , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for B i , i = 1,2,..., B i B j = ,...

Some limit theorems for m -pairwise negative quadrant dependent random variables

Yongfeng Wu, Jiangyan Peng (2018)

Kybernetika

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The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent p ( 1 p 2 ) for m -pairwise negatively quadrant dependent ( m -PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise m -PNQD random variables, which extend and improve the corresponding results in [Y. Meng and Z. Lin (2009)] and [H. S. Sung (2013)]. It is worthy to point out that the open problem of [H. S. Sung, S. Lisawadi, and A. Volodin (2008)] can be...

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...

Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan (2013)

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

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In this paper, we study the size of the giant component C G in the random geometric graph G = G ( n , r n , f ) of n nodes independently distributed each according to a certain density f ( · ) in [ 0 , 1 ] 2 satisfying inf x [ 0 , 1 ] 2 f ( x ) g t ; 0 . If c 1 n r n 2 c 2 log n n for some positive constants c 1 , c 2 and n r n 2 as n , we show that the giant component of G contains at least n - o ( n ) nodes with probability at least 1 - e - β n r n 2 for all n and for some positive constant β . We also obtain estimates on the diameter and number of the non-giant components of G .

Uniform mixing time for random walk on lamplighter graphs

Júlia Komjáthy, Jason Miller, Yuval Peres (2014)

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

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Suppose that 𝒢 is a finite, connected graph and X is a lazy random walk on 𝒢 . The lamplighter chain X associated with X is the random walk on the wreath product 𝒢 = 𝐙 2 𝒢 , the graph whose vertices consist of pairs ( f ̲ , x ) where f is a labeling of the vertices of 𝒢 by elements of 𝐙 2 = { 0 , 1 } and x is a vertex in 𝒢 . There is an edge between ( f ̲ , x ) and ( g ̲ , y ) in 𝒢 if and only if x is adjacent to y in 𝒢 and f z = g z for all z x , y . In each step, X moves from a configuration ( f ̲ , x ) by updating x to y using the transition rule of X and then...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process u d at level u , d 3 . Roughly speaking, we show that observations of u restricted to two disjoint subsets A 1 and A 2 of d are approximately independent, once we add a sprinkling to the process u by slightly increasing the parameter u . Our results differ from previous ones in that we allow the mutual distance between the sets A 1 and A 2 to be much smaller than their diameters. We then provide an important application...

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 + + S n , where S i = X 1 + X 2 + + 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 ) : = max 1 i n S i ( 2 ) l t ; 0 c 𝔼 | S n + 1 | ( n + 1 ) 𝔼 | X 1 | , with c 6 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 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 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 L p whose independent copies span l M in L p is essentially unique.

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 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 ] l i m i n f S / n l i m s u p S / n E * [ X ] , where E * and E* are the lower and upper expectations....

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

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We consider C 2 families t f t of  C 4 unimodal maps f t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure μ t of  f t depends differentiably on  t , as a distribution of order 1 . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of  μ t for a Benedicks-Carleson map f t , in terms of a single smooth function and the...

Weak convergence of mutually independent X B and X A under weak convergence of X X B - X A

W. Szczotka (2006)

Applicationes Mathematicae

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For each n ≥ 1, let v n , k , k 1 and u n , k , k 1 be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let X B ( t ) = ( 1 / c ) j = 1 [ n t ] ( v n , j - v ̅ ) , X A ( t ) = ( 1 / c ) j = 1 [ n t ] ( u n , j - u ̅ ̅ ) , t ≥ 0, and X = X B - X A . The main result gives conditions under which the weak convergence X X , where X is a Lévy process, implies X B X B and X A X A , where X B and X A are mutually independent Lévy processes and X = X B - X A .

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

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Consider a N × n non-centered matrix 𝛴 n with a separable variance profile: 𝛴 n = D n 1 / 2 X n D ˜ n 1 / 2 n + A n . Matrices D n and 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:...

A multiparameter variant of the Salem-Zygmund central limit theorem on lacunary trigonometric series

Mordechay B. Levin (2013)

Colloquium Mathematicae

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We prove the central limit theorem for the multisequence 1 n N 1 n d N d a n , . . . , n d c o s ( 2 π m , A n . . . A d n d x ) where m s , a n , . . . , n d are reals, A , . . . , A d are partially hyperbolic commuting s × s matrices, and x is a uniformly distributed random variable in [ 0 , 1 ] s . The main tool is the S-unit theorem.

Gaussian approximation of Gaussian scale mixtures

Gérard Letac, Hélène Massam (2020)

Kybernetika

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For a given positive random variable V > 0 and a given Z N ( 0 , 1 ) independent of V , we compute the scalar t 0 such that the distance in the L 2 ( ) sense between Z V 1 / 2 and Z t 0 is minimal. We also consider the same problem in several dimensions when V is a random positive definite matrix.

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

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If the space 𝒬 of quadratic forms in n is splitted in a direct sum 𝒬 1 ... 𝒬 k and if X and Y are independent random variables of n , assume that there exist a real number a such that E ( X | X + Y ) = a ( X + Y ) and real distinct numbers b 1 , . . . , b k such that E ( q ( X ) | X + Y ) = b i q ( X + Y ) for any q in 𝒬 i . We prove that this happens only when k = 2 , when n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

Geometric rigidity of × m invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let μ be a probability measure on [ 0 , 1 ] which is invariant and ergodic for T a ( x ) = a x 𝚖𝚘𝚍 1 , and 0 < 𝚍𝚒𝚖 μ < 1 . Let f be a local diffeomorphism on some open set. We show that if E and ( f μ ) E μ E , then f ' ( x ) ± a r : r at μ -a.e. point x f - 1 E . In particular, if g is a piecewise-analytic map preserving μ then there is an open g -invariant set U containing supp μ such that g U is piecewise-linear with slopes which are rational powers of a . In a similar vein, for μ as above, if b is another integer and a , b are not powers of a common integer, and if ν is...

Local density of diffeomorphisms with large centralizers

Christian Bonatti, Sylvain Crovisier, Gioia M. Vago, Amie Wilkinson (2008)

Annales scientifiques de l'École Normale Supérieure

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Given any compact manifold M , we construct a non-empty open subset 𝒪 of the space Diff 1 ( M ) of C 1 -diffeomorphisms and a dense subset 𝒟 𝒪 such that the centralizer of every diffeomorphism in 𝒟 is uncountable, hence non-trivial.

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer k 2 and every set A of words over k satisfying lim sup n | A [ k ] n | / k n > 0 there exist a word c over k and a sequence ( w n ) of left variable words over k such that the set c { c w 0 ( a 0 ) . . . w n ( a n ) : n and a 0 , . . . , a n [ k ] } is contained in A . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne (2017)

Applications of Mathematics

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Acute triangles are defined by having all angles less than π / 2 , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension n 3 , acuteness is defined by demanding that all dihedral angles between ( n - 1 ) -dimensional faces are smaller than π / 2 . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property...