Marjorie G. Hahn

Max-infinitely divisible and max-stable sample continuous processes

SummaryConditions for a process ζ on a compact metric spaceS to be simultaneously max-infinitely divisible and sample continuous are obtained. Although they fall short of a complete characterization of such processes, these conditions yield complete descriptions of the sample continuous non-degenerate max-stable processes onS and of the infinitely divisible non-void random compact subsets of a Banach space under the operation of convex hull of unions.

Central limit theorems in D[0, 1]

SummaryLet X be a stochastic process with sample paths in the usual Skorohod space D[0, 1]. For a sequence {Xn} of independent copies of X, let Sn=X1+⋯+Xn. Conditions which are either necessary or sufficient for the weak convergence of n−1/2(Sn−ESn) to a Gaussian process with sample paths in D[0, 1] are discussed. Stochastically continuous processe are considered separately from those with fixed discontinuities. A bridge between the two is made by a Decomposition central limit theorem.

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.

Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion

In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is the first hitting time process for either a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property.

On q-Gaussians and exchangeability

The q-Gaussian distributions introduced by Tsallis are discussed from the point of view of variance mixtures of normals and exchangeability. For each , there is a q-Gaussian distribution that maximizes the Tsallis entropy under suitable constraints. This paper shows that q-Gaussian random variables can be represented as variance mixtures of normals when q > 1. These variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables, thereby providing a possible model that has been extensively studied in probability theory. The formulation provided has the additional advantage of yielding, for each q, a process which is naturally the q-analog of the Brownian motion. Explicit mixing distributions for q-Gaussians should facilitate applications to areas such as option pricing. The model might provide insight into the study of superstatistics.

SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.

Limit theorems for the logarithm of sample spacings

Many statistical problems can be reformulated in terms of tests of uniformity. Some strong laws of large numbers and a central limit theorem for the logarithm of transformed spacings are obtained. These theorems provide a characterization of the uniform distribution. A general information-type inequality is deduced which gives a quantitative measurement (using the Kullback-Leibler number) of the discrepancy between an arbitrary distribution and the uniform distribution.

Fractional Fokker-Planck-Kolmogorov type equations and their associated stochastic differential equations

There is a well-known relationship between the Itô stochastic differential equations (SDEs) and the associated partial differential equations called Fokker-Planck equations, also called Kolmogorov equations. The Brownian motion plays the role of the basic driving process for SDEs. This paper provides fractional generalizations of the triple relationship between the driving process, corresponding SDEs and deterministic fractional order Fokker-Planck-Kolmogorov type equations.

Affine normability of partial sums of I.I.D. random vectors: A characterization

SummaryLet X, X1,X2,... be i.i.d. d-dimensional random vectors with partial sums Sn. We identify the collection of random vectors X for which there exist non-singular linear operators Tn and vectors υn∈ℝ d such that {ℒ(Tn(Sn−υn)),n>=1} is tight and has only full weak subsequential limits. The proof is constructive, providing a specific sequence {Tn}. The random vector X is said to be in the generalized domain of attraction (GDOA) of a necessarily operator-stable law γ if there exist {Tn} and {υn} such that ℒ(Tn(Sn−υn))→γ. We characterize the GDOA of every operator-stable law, thereby extending previous results of Hahn and Klass; Hudson, Mason, and Veeh; and Jurek. The characterization assumes a particularly nice form in the case of a stable limit. When γ is symmetric stable, all marginals of X must be in the domain of attraction of a stable law. However, if γ is a nonsymmetric stable law then X may be in the GDOA of γ even if no marginal is in the domain of attraction of any law.

On stability of probability laws with univariate stable marginals

SummaryExamples of D. Marcus in ℝ2 dispel the belief that a probability measure on ℝd is stable if and only if all its univariate marginals are stable. However, in ℝd (in fact, in fairly general linear spaces), a probability measure whose two-dimensional marginals are all infinitely divisible is stable if and only if all its univariate marginals are stable.