Arrays Of Independent Random Variables Research Articles

A Central Limit Theorem for incomplete U-statistics over triangular arrays

We analyze the fluctuations of incomplete U-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled and centered version of the U-statistic converges to a normal random variable. Our method of proof relies on a martingale CLT. An application, a CLT for the hitting time for random walks on random graphs, will be presented in Löwe and Terveer (2020).

Limit theorems for ratios of order statistics from uniform distributions

Let {X_{ni}, 1 leq i leq m_{n}, ngeq 1} be an array of independent random variables with uniform distribution on [0, theta _{n}], and {X_{n(k)}, k=1, 2, ldots , m_{n}} be the kth order statistics of the random variables {X_{ni}, 1 leq i leq m_{n}}. We study the limit properties of ratios {R_{nij}=X_{n(j)}/X_{n(i)}, 1leq i < j leq m_{n}} for fixed sample size m_{n}=m based on their moment conditions. For 1=i < j leq m, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for 2=i < j leq m, we obtain some classical limit theorems and self-normalized limit theorems.

Complete convergence for arrays of ratios of order statistics

Abstract Let {Xn,k, 1 ≤ k ≤ mn, n ≥ 1} be an array of independent random variables from the Pareto distribution. Let Xn(k) be the kth largest order statistic from the nth row of the array and set Rn,in,jn = Xn(jn)/Xn(in) where jn &lt; in. The aim of this paper is to study the complete convergence of the ratios {Rn,in,jn, n ≥ 1}.

Discussion on the conditions of logarithmic law for arrays of independent random variables

Logarithmic law; Strong convergence; Array of independent random variables

On the error bound in a combinatorial central limit theorem

Let $\mathbb{X}=\{X_{ij}: 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $\pi$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let $W=\sum_{i=1}^nX_{i\pi(i)}$. Suppose $\mathbb{X}$ is standardized so that ${\mathbb{E}}W=0,\operatorname {Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\sum_{i,j=1}^n{\mathbb{E}}|X_{ij}|^3/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.

Kolmogorov Type Law of the Logarithm for Arrays

In this paper, we establish the analogue of Kolmogorov-type law of the logarithm for an array of independent random variables. The main methods are to develop the classic Bernstein’s and Kolmogorov’s inequality and replace the boundedness condition by conditions on the cumulant transform.

On complete convergence of triangular arrays of independent random variables

Given a triangular array a = { a n , k , 1 ⩽ k ⩽ k n , n ⩾ 1 } of positive reals, we study the complete convergence property of T n = ∑ k = 1 k n a n , k X n , k for triangular arrays X = { X n , k , 1 ⩽ k ⩽ k n , n ⩾ 1 } of independent random variables. In the Gaussian case we obtain a simple characterization of density type. Using Skorohod representation and Gaussian randomization, we then derive sufficient criteria for the case when X n , k are in L p , and establish a link between the L p -case and L 2 p -case in terms of densities. We finally obtain a density type condition in the case of uniformly bounded random variables.

Maxima of the cells of an equiprobable multinomial

Consider a sequence of multinomial random vectors with increasing number of equiprobable cells. We show that if the number of trials increases fast enough, the sequence of maxima of the cells after a suitable centering and scaling converges to the Gumbel distribution. While results are available for maxima of triangular arrays of independent random variables with certain types of distribution, such results in a dependent setup is new. We also prove that the maxima of a triangular sequence of appropriate Binomial random variables have the same limit distribution. An auxiliary large deviation result for multinomial distribution with increasing number of equiprobable cells may also be of independent interest.

A Uniform Bound on a Combinatorial Central Limit Theorem

This paper establishes a combinatorial central limit theorem for an array of independent random variables (X ij ), 1 ≤ i, j ≤ n, (n → ∞) with finite third moments. Let π = (π(1), π(2), …, π(n)) be a permutation of {1, 2, …, n}, and define W n = ∑ i X iπ(i). Then the authors prove the following uniform central limit property: , where F n is the distribution of is the strandrad normal distribution, and with [Xcirc] ij is a suitable normalization of X ij . The proof uses Stein's method and the result generalizes and improves a number of known results.

U-statistics on a lattice of I.I.D. random variables

Partial sums and sample means of r-dimensionally indexed arrays of independent random variables have been studied by Dunford (1951), Zygmund (1951), Kuelbs (1968), Wichura (1969), Smythe (1973), Gut (1978, 1992), Etemadi (1981), Klesov (1981, 1983), and Su and Taylor (1992), whose results cover weak convergence, invariance principles, and almost sure behavior. Applications of such results arise in ergodic theory and the study of Brownian sheets. This paper extends to the case of U-statistics, for example, the sample variance, defined on such an array. An almost sure representation as an i.i.d. average, the central limit theorem for the case of random index, the law of the iterated logarithm, and an invariance principle are developed.

Compound Poisson approximation for extremes of moving minima in arrays of independent random variables

Compound Poisson approximation for extremes of moving minima in arrays of independent random variables

Non-uniform rates of convergence for double arrays of independent random variables with applications

Abstract)Let be a double array of consteds and be a sequence of independent random variables. DenoteIt is well known thatunder some conditions, where is the standard normal distributionfunction.This paper studies the rate of convergence of Fn(x) toΦ(x) depending on n and x wherex is in a neighborhood of the origin under minimal conditions. Explicitly, we establish thefollowing theorems.Theorem 1. Suppose thatReceived January 20, 1994. Revised February 22, 1995.where 0 1. Then, there exist n) b 0 (not depending on n) such that for all n alld allx E R with lxl 5 "L.'/',Theorem 2. Suppose that (4) holds, 0 L. - 0 and t.n E N is a sequence ofconstants with t. ￣ co such that t. ￣ o(L.'/'). ThenRemark 1. Noting that L. S sup la.kl/A. ￣ L;, therefore, Theorems 1 and 2 alsokhold if L. whenever it appears, is replaced by L;.For the particular caseSa reduces to the partial sum Z:=, uk, which has been studied extensively in the literature(see [1,2]). In this cajse, A: = n, L. = .--i. It is well known that Theorem 2 can't beremoved under (4).In time series analysis, an important class of covariance stationary models is defined bywhere {Ek, --co k co) is a sequence of independent random variables and Cd areconstants. The sequence {Z j, j 2 1} is called lillear process generated by {Ek, --co k co}. Since Z:=, Zj = ZZ=--co ankEk where ark = Z:=, Cj--k, making use of Theorems 1and 2 we obtain.Theorem 3. Let (4) hold, Cd be constants such thatThen, there exist n, b 0 such that for all n all x E R with lxl 5 un'/6,Theorem 4. Assume that the conditions of Theorem 3 hold and t. is a sequence ofconstants with t. ￣ co such that t. ￣ o(n'/'). ThenRemark 2. for the following Ck,where the function l(x) 2 0 is slowly varying at infinity, a 2 1 and in or = t, ZC=, k--'l(k) co, Halll61 obtained the rate of convergence of F.(x) to .(x). Applying the same methodas that of dealing with aam ￣ Z:=, sj--k in [6], we can obtain some results similar to (10),(11) for Cd satisfying (12).TO conclude this paper, we give some classical examples (the so-called summabilitymethods) including the most importallt ones from the paint of view of probability (see [3--5]for details. In [51, the authors have dealt with the rate of convergence for SummabilityMethods "lider an assumption of finite third moments).1. Cesaro Sum:2. Abel Sum:3. Borel Sum:4. Euler Sum: (0 p 1)5. Delayed Sum (or RI Sum): (e 0)6. Valiron Sum:7. Taylor Sum: (0 or 1)8. Meyer-K5nig Sum: (0 a 1)

Sums of Independent Triangular Arrays and Extreme Order Statistics

Let $X_{n,i}$ denote an infinitesimal array of independent random variables with convergent partial sums $Z_n = \sum^n_{i=1} X_{n,i} -a_n \rightarrow_\mathscr{D}\xi$. Throughout, we find conditions for the convergence of the portion $k_n$ of lower extremes $L_n(k_n) = \sum^{k_n}_{i=1}X_{i:n} - b_n$ given by order statistics $X_{i:n}$. Similarly, $W_n(r_n)$ denotes the sum of the $r_n$ upper extremes and $M_n = Z_n - L_n - W_n$ stands for the middle part of the sum. It is shown that $(L_n, M_n, W_n) \rightarrow_\mathscr{D} (\xi_1, \xi_2, \xi_3)$ jointly converges for various sequences $k_n, r_n \rightarrow \infty$, where the components of the limit law are independent such that $\xi_1 + \xi_2 + \xi_3 =_\mathscr{D} \xi$. The limit of the middle part $\xi_2$ is asymptotically normal and $\xi_1 (\xi_3)$ gives the negative (positive) spectral Poisson part of $\xi$. In the case of a compound Poisson limit distribution we obtain rates of convergence that can be used for applications to insurance mathematics.

On Iterated Logarithm Laws for Linear Arrays and Nonparametric Regression Estimators

Laws of the iterated logarithm are derived for row sums of triangular arrays of independent random variables, in the context of nonparametric regression estimators. These laws provide exact strong convergence rates for kernel type nonparametric regression estimators. They apply to the important case where design points are conditioned upon, and permit the design to be multivariate. We impose minimal conditions on the error distribution; in fact, only finite variance is needed.

Advanced Probability Theory.

Review of basic concepts expectation and integral, weak and strong convergence transforms of distribution infinite sequences of independent random variables weak convergence triangular arrays of independent random variables, infinitely divisible distributions indipendent and identically distributed random variables conditional exceptions martingales topics in the theory of stochastic processes hints for solutions of exercises.

Limit theorems for triangular arrays under a relaxed asymptotic negligibility condition

AbstractLet {Xnk} be a triangular array of independent random variables satisfying the so-called tail-negligibility condition, i.e. such that Prob{|Xnk| &gt; a} → 0 as both k, n → ∞. It is also assumed that for each fixed k, Xnk converges in distribution as n → ∞. Theorems on the asymptotic behavior of the row sums of the array, analogous to those of the classical theory under the uniform negligibility condition, are presented.

Functional Law of the Iterated Logarithm and Uniform Central Limit Theorem for Partial-Sum Processes Indexed by Sets

Let $\{X_j: \mathbf{j} \in J^d\}$ be an array of independent random variables, where $J^d$ denotes the $d$-dimensional positive integer lattice. The main purpose of this paper is to obtain a functional law of the iterated logarithm (LIL) for suitably normalized and smoothed versions of the partial-sum process $S(B) = \sum_{j \in B}X_j$. The method of proof involves the definition of a set-indexed Brownian process, and the embedding of the partial-sum process in this Brownian process. In addition, the LIL is derived for this Brownian process. The method is extended to yield a uniform central limit theorem for the partial-sum process.

A Law of the Iterated Logarithm for Double Arrays of Independent Random Variables with Applications to Regression and Time Series Models

Motivated by the problem of establishing laws of the iterated logarithm for least squares estimates in regression models and for partial sums of linear processes, we prove a general $\log \log$ law for weighted sums of the form $\sum^\infty_{i=-\infty} a_{ni}\varepsilon_i$, where the $\varepsilon_i$ are independent random variables with zero means and a common variance $\sigma^2$, and $\{a_{ni}: n \geq 1, -\infty < i < \infty\}$ is a double array of constants such that $\sum^\infty_{i=-\infty} a^2_{ni} < \infty$ for every $n$. Besides applying the general theorem to least squares estimates and linear processes, we also use it to improve earlier results in the literature concerning weighted sums of the form $\sum^n_{i=1} f(i/n)\varepsilon_i$.

Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables

Functional Limit Theorems for Extreme Values of Arrays of Independent Random Variables

Laws of the iterated logarithm for nonparametric density estimators

We establish a law of the iterated logarithm for a triangular array of independent random variables, and apply it to obtain laws for a large class of nonparametric density estimators. We consider the case of Rosenblatt-Parzen kernel estimators, trigonometric series estimators and orthogonal polynomial estimators in detail, and point out that our technique has wider application.