Limit Distributions For Sums Research Articles

B.V. Gnedenko: Classic of Limit Theorems in the Theory of Probability

B.V. Gnedenko is an outstanding scientist-mathematician, who worked in the area of probability theory and its applications. B.V. Gnedenko deserved world-wide popularity by his investigations of limit distributions for sums of independent random variables (Gnedenko and Kolmogorov 1954) that completed a long period of probability theory development up to the middle of XX-th century. The main idea of “accompanied infinitely divisible distributions” developed by B.V. Gnedenko, became a guidance in the limit theory of semimartingales as it is presented by Jacod and Shiryaev (1987) and others (Cinlar et al. 1980). The triplet of predictable characteristics for semimartingale is the main idea in investigation the limit behavior for the random evolutions in the scheme of Poisson approximation (Koroliuk and Limnios, Theory Probab Appl 49(4):629–644, 2005b).

Nonstationary INAR(1) Process with th-Order Autocorrelation Innovation

This paper is concerned with an integer-valued random walk process withqth-order autocorrelation. Some limit distributions of sums about the nonstationary process are obtained. The limit distribution of conditional least squares estimators of the autoregressive coefficient in an auxiliary regression process is derived. The performance of the autoregressive coefficient estimators is assessed through the Monte Carlo simulations.

Pfeifer records process

In this article, we study the limit distribution of sums of Pfeifer records. Motivated by the results obtained by Arnold and Villaseñor [Generalized order statistics process and Pfeifer records, Statistics 46(3) (2012), pp. 373–385], we show that the partial sum process of Pfeifer records converge to a function of the Brownian motion. The normalization is either a sequence of appropriate constants or a sequence of functions, depending on the tail behaviour of the underlying variables. These results, in particular, prove stronger version of results obtained in Villaseñor and Arnold [On limit laws for sums of Pfeifer records, Extremes 10 (2007), pp. 235–248] and Bose and Gangopadhyay [Convergence of linear functions of Pfeifer records, Extremes 13 (2010), pp. 89–106] and extends results of Bose et al. [Partial sum process for records, Extremes 8 (2005), pp. 43–56] from classical records to Pfeifer records.

Product-Convolution of R-Symmetric Unimodal Distributions: An Analogue of Wintner's Theorem

Abstract Gnedenko and Kolmogorov (1949) in their acclaimed monograph. Limit Distributions for Sums of Independent Random Variables, claimed that the convolutions of unimodal distributions are unimodal. Kai Lai Chung, in an appendix of his English translation of the monograph, by a counterexample, refuted the claim and further noted Wintner's (1938) result that the convolutions of symmetric unimodal distributions are symmetric unimodal. In this note, it is shown that the product-convolutions of unimodal distributions are not unimodal either. Furthermore, an analogue of Wintner's result based on the relatively recent notion of R-symmetry (Mudholkar and Wang, 2007) is offered by showing that the product-convolutions of R-symmetric unimodal distributions are R-symmetric unimodal.

On limit laws for sums of Pfeifer records

The problem of determining limiting distributions for sums of records has been studied by several authors who have considered a variety of assumptions sufficient to ensure that sums of records properly normalized will converge to a non-degenerate distribution. As a parallel to these endeavors, it is of interest to establish conditions under which the sum of Pfeifer records, properly normalized, converges. Pfeifer records are defined under the assumption that initial observations are i.i.d. with common survival function \(\left( \ \overline{F_{0}}\right) ^{\alpha _{0}}\) and following the (n−1)-th record value the observations are assumed to have survival function \(\left( \ \overline{F_{0}}\right) ^{\alpha _{n}}\) ,n=1,2,.... The study of the asymptotic behavior of sums of Pfeifer records constitutes a natural generalization of work on sums of classical records. The present paper introduces conditions under which the limit distribution of sums of Pfeifer records is non-degenerate.

Randomly fractionally integrated processes

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (dt)t∈ℤ of real numbers; if the parameter sequence is constant dt ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d. They also studied partial sums limits of filtered white noise nonstationary processes A(d)et and B(d)et for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes XtA = A(d)et and XtB = B(d)et by assuming that d = (dt, t ∈ ℤ) is a random iid sequence, independent of the noise (et). In the case where the mean \(\bar d = \mathbb{E}d_0 \in \left( {0,1/2} \right)\), we show that large sample properties of XA and XB are similar to FARIMA(0, \(\bar d\), 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter \(\bar d + (1/2)\). The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(XtA) of a randomly fractionally integrated process XtA with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence dt, this reduces to the standard Hermite rank used in Dobrushin and Major [2].

On Infinitely Divisible Distributions on Locally Compact Abelian Groups

Our aim in this paper is to characterize some classes of infinitely divisible distributions on locally compact abelian groups. Firstly infinitely divisible distributions with no idempotent factor on locally compact abelian groups are characterized by means of limit distributions of sums of independent random variables. We introduce semi-selfdecomposable distributions on topological fields, and in case of totally disconnected fields we give a limit theorem for them. We also give a characterization of semistable laws on p-adic field and show that semistable processes are constructed as scaling limits of sums of i.i.d.

Limit distributions of sums of random variables defined on a finite homogeneous Markov chain. I

Limit distributions of sums of random variables defined on a finite homogeneous Markov chain. I

On Classes of Limit Distributions for Sums of a Random Number of Identically Distributed Independent Random Variables

On Classes of Limit Distributions for Sums of a Random Number of Identically Distributed Independent Random Variables

Limit Distributions for Sums of Independent Random Variables.

Limit Distributions for Sums of Independent Random Variables.

On Limit Distributions of Sums of Conditionally Independent Random Variables

This paper deals with limit distributions for sums $\eta _n $ which become independent when a certain path $x_n $, $n = 0,1,2, \cdots $, of a Markov chain is defined. The dependence between $\{ \eta _n \}$ and $\{ X_n \}$ is expressed more exactly by (1). Let $X_s $ be the path of a continuous Markov process. Furthermore, the study of the limit distributions of $\xi (t) = \int_0^t f ( X_s )ds$ at $t \to \infty $ can be reduced to the study of limit distributions of sums $\eta _n $. This reduction is illustrated for the case where $X_s $ is a one-dimensional diffusion process. The limit distribution for $\xi (t)$ coincides with distributions obtained in [12]. The sufficient conditions for convergence to each distribution are also given (Theorems 2 and 3).

Limit Distributions for Sums of Independent Random Variables. By B.V. Gnedenko and A.N. Kolmogorov. Translated by K.L. Chung. Pp. ix, 264. $7.50. 1954. (Addison-Wesley, Cambridge, Mass.)

Limit Distributions for Sums of Independent Random Variables. By B.V. Gnedenko and A.N. Kolmogorov. Translated by K.L. Chung. Pp. ix, 264. $7.50. 1954. (Addison-Wesley, Cambridge, Mass.) - Volume 39 Issue 330

Limit Distributions for Sums of Independent Random Variables.

Limit Distributions for Sums of Independent Random Variables.

Limit Distributions for Sums of Independent Variables

Limit Distributions for Sums of Independent Variables

Limit Distributions for Sums of Independent Random Variables.

Limit Distributions for Sums of Independent Random Variables.