A limit theorem for the moments of sums of independent random variables

Forn≧1, letS _n=ΣX _n,i (1≦i≦r _n <∞), where the summands ofS _n are independent random variables having medians bounded in absolute value by a finite number which is independent ofn. Letf be a nonnegative function on (− ∞, ∞) which vanishes and is continuous at the origin, and which satisfies, for some for allt≧1 and all values ofx. Theorem.For centering constants c _n,let S _n − c _n converge in distribution to a random variable S. (A)In order that Ef(S_n − c_n) converge to a limit L, it is necessary and sufficient that there exist a common limit (B)If L exists, then L<∞ if and only if R<∞, and when L is finite, L=Ef(S)+R. Applications are given to infinite series of independent random variables, and to normed sums of independent, identically distributed random variables.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sumsa.ud their maxima. In this paper, we generalize their work to the situation thatis a sequence of upper tail asymptotically independent random variables with common distribution from the is a sequence of nonnegative random variables, independent of and satisfying some regular conditions. Moreover. no additional assumption is required on the dependence structureof {θi,i≥ 1）.     

Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^＋, {N（n）; n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C （0, 1）, n ≥ 0, let {M（n）; n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N（n）; n ≥ 0}, {Xn; n≥1} and {M（n）; n ≥ 0} are mutually independent. Write S（n） =N（n）∑i=1 Xi-cM（n）.Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.     

Matrix Measures, Random Moments, and Gaussian Ensembles

We consider the moment space M_n\mathcal{M}_{n} corresponding to p×p real or complex matrix measures defined on the interval [0,1]. The asymptotic properties of the first k components of a uniformly distributed vector (S_1,n, ... , S_n,n)^* ~ U (M_n)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S _1,n ,…,S _k,n )^∗ converges weakly to a vector of k independent p×p Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first k canonical moments corresponding to the uniform distribution on the real or complex moment space M_n\mathcal{M}_{n} are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.     

On the strong convergence of weighted sums

Let {X _n ,n≥1} be a sequence of independent and identically distributed random variables and {a _ni ,1≤i≤n,n≥1} an array of constants. Some strong convergence results for the weighted sums ∑ _i=1 ⁿ a _ni X _i are obtained.     

On the range of random walk

Let {S _n , n=0, 1, 2, …} be a random walk (S _n being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE _d , thed-dimensional integer lattice. Letf _n =Prob {S ₁ ≠ 0, …,S _n −1 ≠ 0,S _n =0 |S ₀=0}. The random walk is said to be transient if and strongly transient if . LetR _n =cardinality of the set {S ₀,S ₁, …,S _n }. It is shown that for a strongly transient random walk with p<1, the distribution of [R _n −np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S ₀, …,S _n }. For a finite setA inE _d , let C(A=Σ _x∈A ) Prob {S _n ∉A, n≧1 |S ₀=x} be the capacity ofA. A strong law forC{S ₀, …,S _n } is proved for a transient random walk, and some related questions are also considered. This research was partially supported by the National Science Foundation.     

A Theorem related to Marcinkiewicz-Salem Conjecture

Let S_n, n = 1,2… be the sequence of partial sums of independent Bernoulli random variables. We show, for the randomly sampled trigonometric system {e_S?,? in N}, the validity of the Marcinkiewicz-Salem Conjecture.     

The average of the values of a function at random points

The behavior of (1/N) asN→∞ is considered, wheref is a bounded measurable function on (−∞, ∞) and (S _n) _n ^=1/∞ are the partial sums of a sequence of independent and identically distributed rondom variables.     

General Weak Laws of Large Numbers for Bootstrap Sample Means

Abstract

For bootstrap sample means resulting from a sequence {X _n , n ≥ 1} of random variables, very general weak laws of large numbers are established. The random variables {X _n , n ≥ 1} do not need to be independent or identically distributed or be of any particular dependence structure. In general, no moment conditions are imposed on the {X _n , n ≥ 1}. Examples are provided that illustrate the sharpness of the main results.     

Some limit theorems for negatively associated random variables

Let {X _n , n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L ^p -convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.     

Berry–Esseen Bound for a Sample Sum from a Finite Set of Independent Random Variables

Let {X ₁,...,X _N} be a set of N independent random variables, and let S _n be a sum of n random variables chosen without replacement from the set {X ₁,...,X _N} with equal probabilities. In this paper we give an estimate of the remainder term for the normal approximation of S _n under mild conditions.     

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent(NOD) random variables are investigated. Let {X_n, n ≥ 1}be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.     

Uniformly Accurate Quantile Bounds for Sums of Arbitrary Independent Random Variables

Fix any n≥1. Let X ₁,…,X _n be independent random variables such that S _n =X ₁+⋅⋅⋅+X _n , and let S^*_n=sup_{1 ￡ k ￡ n}S_kS^{*}_{n}=\sup_{1\le k\le n}S_{k} . We construct upper and lower bounds for s _y and s_y^*s_{y}^{*} , the upper \frac1y\frac{1}{y} th quantiles of S _n and S^*_nS^{*}_{n} , respectively. Our approximations rely on a computable quantity Q _y and an explicit universal constant γ _y , the latter depending only on y, for which we prove that

Complete convergence and almost sure convergence of weighted sums of random variables

Letr>1. For eachn1, let {X _nk , –<k<} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee for every >0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.     

Laws of large numbers for weighted sums of random variables and random elements

Consider the weighted sums of a sequence {X _n} of independent random variables or random elements inD [0,1]. For convergence ofS _n in probability and with probability one, in [2],[3] etc., the following stronger condition is required: {X _n} is uniformly bounded by a random variableX,i.e.P(¦X _n¦x)P(¦X¦x) for allx>0. Our paper aims at trying to drop this restriction.The Project supported by National Natural Science Foundation of China     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

Large deviations for sums of independent random variables with dominatedly varying tails

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.     

The functional law of the iterated logarithm for truncated sums

The functional law of the iterated logarithm (FLIL) is obtained for truncated sums $S_n = \sum _{j = l}^n X_j I\{ X_j^{\text{2}} \leqslant b_n \} $ of independent symmetric random variables X_j, 1<-j≤n, b_n≤∞. Considering the random normalization $T_n^{1/{\text{2}}} = \left( {\sum\limits_{j = 1}^n {X_j^{\text{2}} } I\{ X_j^{\text{2}} \leqslant b_n \} } \right)^{1/{\text{2}}} ,$ we obtain an upper estimate in the FLIL, using only the condition that T_n→∞ almost surely. These results are useful in studying trimmed sums. Bibliography: 9 titles.     

Two identities for sums of lattice random variables

For the sums S_n of independent, identically distributed random variables X_k, assuming nonnegative integer values, if EX₁ < 1/m,=">