Asymptotic Behavior Of Partial Sums Research Articles

Strong approximations for long memory sequences based partial sums, counting and their Vervaat processes

We study the asymptotic behaviour of partial sums of long range dependent random variables and that of their counting process, together with an appropriately normalized integral process of the sum of these two processes, the so-called Vervaat process. The first two of these processes are approximated by an appropriately constructed fractional Brownian motion, while the Vervaat process in turn is approximated by the square of the same fractional Brownian motion.

Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Limit Theorems for Long-Memory Stochastic Volatility Models with Infinite Variance: Partial Sums and Sample Covariances

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Non-central limit theorems for random selections

Selection from finite sets is a basic procedure of statistics and the partial sum behavior of selected elements is completely known under the “uniform asymptotic negligibility” condition of central limit theory. The purpose of the present paper is to determine the asymptotic behavior of partial sums when the central limit theorem fails. As an application, we describe the limiting properties of permutation and bootstrap statistics in case of infinite variance.

On linear processes with dependent innovations

We consider asymptotic behavior of partial sums and sample covariances for linear processes whose innovations are dependent. Central limit theorems and invariance principles are established under fairly mild conditions. Our results go beyond earlier ones by allowing a quite wide class of innovations which includes many important nonlinear time series models. Applications to linear processes with GARCH innovations and other nonlinear time series models are discussed.

A Multiplicative Analogue of Schur's Tauberian Theorem

AbstractA theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

Partial sums of hypergeometric series of unit argument

The asymptotic behaviour of partial sums of generalized hypergeometric series of unit argument is investigated.

Asymptotic behavior of partial sums of a series of probabilities of large deviations

Let {Xk} k=1∞ be a sequence of independent, symmetric random variables with characteristic functions fk(t), The asymptotic behavior of the sum (for arbitrary ɛ > 0) is investigated under the assumption that fk(t) belong to the domain of attraction of a stable law with exponent α (0 < α ≤ 2).

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Summary. By the use of conditioning, we extend previously obtained results on the asymptotic behavior of partial sums for certain triangular arrays of dependent random variables, known as Curie-Weiss models. These models arise naturally in statistical mechanics. The relation of these results to multiple phases, metastable states, and other physical phenomena is explained.

Verteilungs-invarianzprinzipien f�r das starke gesetz der gro\en zahl

The asymptotic behaviour of the stochastic process $$k \to \frac{1}{k}\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}k} {X_i }$$ as k→∞-X ={X i : i=1,2,⋯ being a sequence of independent random variables having mean 0 and positive finite variance, satisfying both Lindeberg's condition and the strong law of large numbers — is studied by means of a distribution invariance principle. This invariance principle sharpens the classical one due to Donsker and Prokhorov describing the “weak” asymptotic behaviour of partial sums of independent random variables on a semi-infinite time interval. The topology of the path space being appropriately chosen it allows to compute the limit distributions of certain functionals associated to X, such as $$X \to \left( {\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}n} {EX_i^2 } } \right)^{1/2} \mathop {\max }\limits_{k{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } {\text{ }}n} \frac{1}{k}\left| {\sum\limits_{i{\text{ }}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } {\text{ }}k} {X_i } } \right|{\text{ (}}n \to \infty {\text{)}}{\text{.}}$$ Moreover, for uniformly bounded variables X i , a general estimate of the rapidity of convergence is derived and applied to various special cases