Non-negative Integer-valued Random Variables Research Articles

On Strong Deviation Theorems Concerning Arrays of Non-Negative Integer-Valued Random Variables

On Strong Deviation Theorems Concerning Arrays of Non-Negative Integer-Valued Random Variables

Poisson approximation in terms of the Gini–Kantorovich distance

It is long known that the distribution of a sum Sn of independent non-negative integer-valued random variables can often be approximated by a Poisson law: Sn≈πλ, where . The problem of evaluating the accuracy of such approximation has attracted a lot of attention in the past six decades. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, it provides insights into Kolmogorov’s problem. Among popular metrics considered in the literature is the Gini–Kantorovich distance dG. The task of establishing an estimate of dG(Sn;πλ) with correct (the best possible) constant at the leading term remained open for a long while. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We show that the accuracy of approximation can be considerably better if the random variables obey an extra condition involving the first two moments. A sharp estimate of the accuracy of shifted (translated) Poisson approximation is established as well.

Discrete line integral on uniform grids: Probabilistic interpretation and applications

Following the methodology developed by (Comput. Math. Appl. 33 (1997) 81–104), we define a discrete version of gradient vector and associated line integral along arbitrary path connecting two nodes of uniform grid. An exponential representation of joint survival function of bivariate discrete non-negative integer-valued random variables in terms of discrete line integral is established. We apply it to generate a discrete analogue of the Sibuya-type aging property, incorporating many classical and new bivariate discrete models. Several characterizations and closure properties of this class of bivariate discrete distributions are presented.

Poisson Approximation for Call Function via Stein–Chen Method

Let V be a sums of independent nonnegative integer-valued random variables and $$h_z$$ be a call function defined by $$h_{z}(v)=(v-z)^+$$ for $$v\ge 0, z \ge 0$$ where $$(v-z)^+=\max \{v-z,0\}$$. In this paper, we give bounds of Poisson approximation for $${E}[h_{z}(V)]$$. These bounds improve the results of Jiao and Karoui (Finance Stoch 13(2):151–180, 2009). The technique used is Stein–Chen method with the zero bias transformation. One example of applications for a call function in finance is the standard collateralized debt obligation (CDO) tranche pricing. The CDO is a security backed by a diversified pool of debt obligation such as bounds, loans and credit default swaps.

On large deviations for sums of discrete m-dependent random variables

ABSTRACTThe ratio is investigated for three cases: (a) when is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and is Poisson rv; (b) when is a statistic of 2-runs and is negative binomial rv; and (c) when is a statistic of -events and is a binomial rv. We also consider the approximation of by the Poisson distribution with parameter depending on x.

The max-BARMA models for counts with bounded support

In this note, we introduce a discrete counterpart of the conventional max-autoregressive moving-average process of Davis and Resnick (1989), based on the binomial thinning operator and driven by a sequence of i. i. d. nonnegative integer-valued random variables with a finite range of counts. Basic probabilistic and statistical properties of this new class of models are discussed in detail, namely the existence of a stationary distribution, and how observations’ and innovations’ distributions are related to each other. Furthermore, parameter estimation is also addressed.

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

ABSTRACTUndergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.

A factorial Moment Distance and an Application to the Matching Problem

In this note we introduce the notion of factorial moment distance for nonnegative integer-valued random variables, and we compare it with the total variation distance. Furthermore, we study the rat...

The max-INAR(1) model for count processes

This paper proposes a discrete counterpart of the conventional max-autoregressive process of order one. It is based on the so-called binomial thinning operator and driven by a sequence of independent and identically distributed nonnegative integer-valued random variables with either regularly varying right tail or exponential-type right tail. Basic probabilistic and statistical properties of the process are discussed in detail, including the analysis of conditional moments, transition probabilities, the existence and uniqueness of a stationary distribution, and the relationship between the observations’ and innovations’ distribution. We also provide conditions on the marginal distribution of the process to ensure that the innovations’ distribution exists and is well defined. Several examples of families of distributions satisfying such conditions are presented, but also some counterexamples are analyzed. Furthermore, the analysis of its extremal behavior is also considered. In particular, we look at the limiting distribution of sample maxima and its corresponding extremal index.

Asymptotics of conditional probabilities of succesful allocation of random number of particles into cells

Abstract The article is devoted to the memory of Valentin Fedorovich Kolchin. Let ζ, ζi (i ∈ N) be independent identically distributed nonnegative integer-valued random variables, (η i1,…, ηiN ) be the fillings of cells in the generalized scheme of allocation of ζi particles into N cells, 1 ≤ i ≤ n, for fixed Zn = (ζ 1, …, ζn ) these allocation schemes are independent. We consider the conditional probabilities P(A n,N | Zn ) of the event A n,N = {each cell in each of n allocation schemes contains no more than r particles}, where r is some fixed number. The sufficient conditions for the convergence of the sequence P(A n,N | Zn ) to a nonrandom limit with probability 1 are given. It is shown that the random variable ln P(A n,N | Zn ) is asymptotically normal. Applications of the obtained results to the noise-proof encoding are discussed.

Infinitely Divisible Approximations for Sums of m-Dependent Random Variables

Assuming conditions on factorial cumulants, we estimate the closeness of distribution of a sum of nonnegative integer-valued m-dependent random variables to the class of all infinitely divisible laws. The accuracy of approximation is measured in total variation and local metrics. Our results are exemplified by an analogue of the first uniform Kolmogorov theorem for the statistic of \((k_1,k_2)\) events.

A factorial moment distance and an application to the matching problem

В этой заметке мы вводим понятие расстояния факториальных моментов для неотрицательных целочисленных случайных величин и сравниваем его с расстоянием по вариации. Кроме этого, мы изучаем скорость сходимости в классической задаче о составлении пар и в обобщенной задаче о составлении пар.

AN EXTENSION OF POISSON APPROXIMATION BY $w$-FUNCTIONS

We extend the result of Poisson approximation (4) to approximate the distribution of a sum of independent non-negative integer-valued random variables. A general upper bound for the total variation distance in terms of each w-function is obtained. Examples have been given to illustrate applications of the result.

A Comparison of the Moment and Factorial Moment Bounds for Discrete Random Variables

In this note, we establish the superiority of the factorial moment bound over the moment bound for nonnegative integer-valued discrete random variables. This solves a problem given earlier. We use some results from approximation theory/ numerical analysis to compare the bounds.

Approximation for expectations of unbounded functions of dependent integer-valued random variables

Expectations of unbounded functions of dependent nonnegative integer-valued random variables are approximated by the expectations of the functions of independent copies of these random variables. The Lindeberg method is used.

Approximation for expectations of unbounded functions of dependent integer-valued random variables

Expectations of unbounded functions of dependent nonnegative integer-valued random variables are approximated by the expectations of the functions of independent copies of these random variables. The Lindeberg method is used.

On construction and simulation of count data models

The mean–variance range and the shape flexibility are important measures of the applicability of a count data model. This paper develops a method for constructing nonnegative integer-valued random variables with any interval domain, any theoretically possible mean–variance pair, and different shapes. The basic tool is a simple mean-preserving discretization procedure for random variables. Two corresponding variate generation algorithms are derived, and shown to be comparable to the alias method. As an application, our method enables production of count data models with full under- and over-dispersion flexibility and desired shape.

On the transition of distributions of sums of random variables related to the generalised allocation scheme from one lattice to another

We consider in detail the phenomenon of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables related to the generalised allocation scheme from one lattice to another.

On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme

We consider the effect of transition of distributions of sums of independent identically distributed non-negative integer-valued random variables from one lattice to another in the framework of the generalised allocation scheme.

Large Deviations for Sums of Independent Heavy-Tailed Random Variables

We obtain precise large deviations for heavy-tailed random sums $$S(t) = \sum\nolimits_{i = 1}^{N(t)} {X_i ,t \geqslant 0} $$ , of independent random variables. $$(N(t))_{t \geqslant 0} $$ are nonnegative integer-valued random variables independent of r.v. (X i )i $$ \in $$ N with distribution functions F i. We assume that the average of right tails of distribution functions F i is equivalent to some distribution function with regularly varying tail. An example with the Pareto law as the limit function is given.