Central Limit Theorem For Products Research Articles

The almost sure local central limit theorem for products of partial sums under negative association

Let {X_{n}, ngeq1} be a strictly stationary negatively associated sequence of positive random variables with mathrm{E}X_{1}=mu>0 and operatorname{Var}(X_{1})=sigma^{2}<infty. Denote S_{n}=sum_{i=1}^{n}X_{i}, p_{k}=mathrm{P}(a_{k}leq ({prod}_{j=1}^{k}S_{j}/(k!mu^{k}) )^{1/(gammasigma_{1} sqrt{k})}< b_{k}) and gamma=sigma/mu the coefficient of variation. Under some suitable conditions, we derive the almost sure local central limit theorem \t\t\tlimn→∞1logn∑k=1n1kpkI{ak≤(∏j=1kSjk!μk)1/(γσ1k)<bk}=1a.s.,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\lim_{n\\rightarrow\\infty}\\frac{1}{\\log n}\\sum_{k=1}^{n} \\frac{1}{kp_{k}}\\mathrm{I} \\biggl\\{ a_{k}\\leq \\biggl(\\frac {\\prod_{j=1}^{k}S_{j}}{k!\\mu^{k}} \\biggr)^{1/(\\gamma\\sigma_{1} \\sqrt {k})}< b_{k} \\biggr\\} =1 \\quad\\mbox{a.s.,} $$\\end{document} where sigma_{1}^{2}=1+frac{1}{sigma^{2}}sum_{j=2}^{infty}operatorname{Cov}(X_{1},X_{j})>0.

A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix $T_0$. Focusing on the eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required $T_0$ to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schr\"odinger operators we can improve some result by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular we solve a problem posed in their work.

A universal result in almost sure central limit theorem for products of sums of partial sums under mixing sequence

Let be a strictly stationary mixing sequence of positive random variables with and Denote and the coefficient of variation. Under some suitable conditions, the author shows that a universal version of almost sure central limit theorem for products of sums of partial sums holds under the assumptions that the mixing coefficients satisfy , moreover we no longer restrict to logarithmic averages, but allow rather arbitrary weight sequences.

Almost sure central limit theorem for products of sums of partial sums

Considering a sequence of i.i.d. positive random variables, for products of sums of partial sums we establish an almost sure central limit theorem, which holds for some class of unbounded measurable functions.

An Extension of the Almost Sure Central Limit Theorem for Products of Sums Under Association

We show that if {X n , n ≥ 1} is a strictly stationary positively or negatively associated sequence of positive random variables with , and (d k ) is a positive numerical sequence obeying a condition similar to Kolmogorov's condition for the LIL, then letting we have where , γ = σ/μ the coefficient of variation, F(·) is the distribution function of the random variable , and 𝒩 is a standard normal random variable. This shows that logarithmic means, used traditionally in a.s. central limit theory for products of sums, can be replaced by other means, leading to considerably sharper results.

A note on the ASCLT for triangular arrays of random variables with an extension to U-statistics

In this paper, we obtain an almost sure central limit theorem for products of independent sums of positive random variables. An extension of the result gives an ASCLT for the U-statistics.

Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables

We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables.

An almost sure central limit theorem for products of sums of partial sums under association

Let { X n , n ⩾ 1 } be a strictly stationary positively or negatively associated sequence of positive random variables with E X 1 = μ > 0 , and Var X 1 = σ 2 < ∞ . Denote S n = ∑ i = 1 n X i , T n = ∑ i = 1 n S i and γ = σ / μ the coefficient of variation. Under suitable conditions, we show that ∀ x lim n → ∞ 1 log n ∑ k = 1 n 1 k I { ( 2 k ∏ j = 1 k T j k ! ( k + 1 ) ! μ k ) 1 / ( γ σ 1 k ) ⩽ x } = F ( x ) a.s. , where σ 1 2 = 1 + 2 σ 2 ∑ j = 2 ∞ Cov ( X 1 , X j ) , F ( ⋅ ) is the distribution function of the random variables e 10 / 3 N and N is a standard normal random variable.

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.

An almost sure central limit theorem for products of sums under association

Let { X n , n ⩾ 1 } be a strictly stationary positively or negatively associated sequence of positive random variables with E X 1 = μ > 0 and Var ( X 1 ) = σ 2 < ∞ . Denote S n = ∑ i = 1 n X i and γ = σ / μ the coefficient of variation. Under suitable conditions, we show that ∀ x lim n → ∞ 1 log n ∑ k = 1 n 1 k I ∏ j = 1 k S j k ! μ k 1 / ( γ σ 1 k ) ⩽ x = F ( x ) a . s ., where σ 1 2 = 1 + 2 σ 2 ∑ j = 2 ∞ Cov ( X 1 , X j ) , F ( · ) is the distribution function of the random variable e 2 N , and N is a standard normal random variable. This extends the earlier work on independent, positive random variables (see Khurelbaatar and Rempala [2006. A note on the almost sure limit theorem for the product of partial sums. Appl. Math. Lett. 19, 191–196]).

Diffusion approximations for random walks on nilpotent Lie groups

We prove a central limit theorem for products of centered i.i.d. r.v.'s taking values on nilpotent, graded, simply connected Lie groups. The main tool is given by weak approximation results for SDEs.

Zariski closure and the dimension of the Gaussian law of the product of random matrices. I

We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG. In this article we present a detailed exposition of results announced in [GGu]. For reasons explained in the introduction, this part is devoted to the case ofSL(m, ℝ) group. The general semi-simple Lie group will be considered in the second part of the work. The central limit theorem for products of independent random matrices is our main topic, and the results obtained complete to a large extent the general picture of the subject. The proofs rely on methods from two theories. One is the theory of asymptotic behaviour of products of random matrices itself. As usual, the existence of distinct Lyapunov exponents is the most important fact here. The other is the theory of algebraic groups. We want to point out that algebraic language and methods play a very important role in this paper. In fact, this mixture of methods has already been used for the study of Lyapunov exponents in [GM1, GM2, GR3]. We believe that it is impossible to avoid the algebraic approach if one aims to obtain complete and effective answers to natural problems arising in the theory of products of random matrices. In order also to present the general picture of the subject we describe several results which are well known. Some of these can be proven for stationary sequences of matrices, others are true also for infinite dimensional operators (see e.g. [BL, O, GM2, L, R]). But our main concern is with independent matrices, in which case very precise and constructive statements can be obtained.

Zariski closure and the dimension of the Gaussian low of the product of random matrices. I

We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG.

Central limit theorems for random evolutions

A product formula for linear operators is used to get a central limit theorem for products of dependent random linear and nonlinear operators on R d driven by a continuous time Markov chain.

A Central Limit Theorem for products of dependent random linear and nonlinear operators

Using a semigroup product formula similar to that of P. Chernoff's, we generalize Marc Berger's Central Limit Theorem for products of independent random matrices to a Central Limit Theorem for products of dependent random linear and nonlinear operators.

Ideal metrics and products of independent random variables

A new class of metrics is introduced, and by studying their properties it is possible to obtain bounds for the remainder in a central limit theorem for products of independent random variables.

Central Limit Theorem for Products of Random Matrices

Using the semigroup product formula of $\text {P}$. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.

Central limit theorem for products of random matrices

Using the semigroup product formula of P \text {P} . Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.