Daniel C. Weiner

Sums, trimmed sums and extremes

I Approaches to Trimming and Self-normalization Based on Analytic Methods.- Asymptotic Behavior of Partial Sums: A More Robust Approach Via Trimming and Self-Normalization.- Weak Convergence of Trimmed Sums.- Invariance Principles and Self-Normalizations for Sums Trimmed According to Choice of Influence Function.- On Joint Estimation of an Exponent of Regular Variation and an Asymmetry Parameter for Tail Distributions.- Center, Scale and Asymptotic Normality for Censored Sums of Independent, Nonidentically Distributed Random Variables.- A Review of Some Asymptotic Properties of Trimmed Sums of Multivariate Data.- II The Quantile-Transform-Empirical-Process Approach to Trimming.- The Quantile-Transform-Empirical-Process Approach to Limit Theorems for Sums of Order Statistics.- A Note on Weighted Approximations to the Uniform Empirical and Quantile Processes.- Limit Theorems for the Petersburg Game.- A Probabilistic Approach to the Tails of Infinitely Divisible Laws.- The Quantile-Transform Approach to the Asymptotic Distribution of Modulus Trimmed Sums.- On the Asymptotic Behavior of Sums of Order Statistics from a Distribution with a Slowly Varying Upper Tail.- Limit Results for Linear Combinations.- Non-Normality of a Class of Random Variables.

Moments of distributions attracted to operator-stable laws

Bounds on the norming operators for distributions in the domain of attraction of an operator-stable distribution are found. These bounds are used to establish the existence and nonexistence of moments of distributions in the domain of attraction of an operator-stable distribution. Similar results for stochastically compact sequences are obtained.

The asymptotic distribution of magnitude-Winsorized sums via self-normalization

Asymptotic normality, tightness, and weak convergence of the magnitude-Winsorized sums formed from symmetric i.i.d. random variables are studied via a new approach that first derives self-normalized (“studentized”) results and then uses these to derive results for constant normalizations. An application of this method to trimmed sums is also discussed to demonstrate its more general applicability as well as to illustrate its use.

Asymptotic Behavior of Partial Sums: A More Robust Approach Via Trimming and Self-Normalization

If X1, X2, X3, ...,are independent, identically distributed (i.i.d.) random variables and \(S_n = \sum\nolimits_{i = 1}^n {X_i }\) otes the nth partial sum, then limit theorems such as the law of large numbers (LLN), the central limit theorem (CLT), and the law of the iterated logarithm (LIL) all involve a strong interplay between the maximal terms of the sample.{|X 1|,...,|X n|} and the asymptotic behavior of the partial sum S n . Indeed, what is shown in the proofs of the classical formulation of each of these results is that the maximal or extreme terms of the sample are negligible in a sense required for the corresponding theorem. Furthermore, since the assumptions sufficient to prove the classical version of each of these results are also necessary, we see that extensions of these limit theorems will likely require methods that nullify or at least limit the effect of the extreme terms.

On Joint Estimation of an Exponent of Regular Variation and an Asymmetry Parameter for Tail Distributions

A random variable X is said to have a joint tail distribution which is regularly varying of index -α if for each c > 0,

Center, Scale and Asymptotic Normality for Censored Sums of Independent, Nonidentically Distributed Random Variables

\mathop {\lim }\limits_{t \to \infty } \frac{{P(\left| X \right| > \,ct)}}{{P(|X|\, > \,t)}}\, = \,c^{ - \alpha }.

Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits II

This paper investigates the related properties of center and scale for distributions and features application to asymptotic normality criteria for sums of independent random variables. Necessary and sufficient conditions are given for convergence in distribution to the standard normal for suitably centered and scaled sums of independent variables in the context of uniform asymptotic negligibility of the summands (Theorem 5.1). New here is the expression of the precise conditions and prior construction of centering and scaling constants directly in terms of the distributions of the original variables, without recourse to symmetrization, and the display of the centered, scaled sums as sums of uniformly asymptotically negligible summands without the necessity of further recentering (cf. Gnedenko and Kolmogorov (1968), Chapter 6, Theorem 2, and Hahn and Klass (1981), Theorem 2 and especially Remark 2 following Theorem 1). The resulting theorem allows a direct and informative expression of Levy’s (1937) profound realization of asymptotic normality for sums as equivalent to negligibility of the summand of maximum modulus (cf. Remark 5 and equation (5.9) here following the statement of Theorem 5.1).