Sequence Of Integer-valued Random Variables Research Articles

A Note on the Accuracy of Normal Approximation of Random Quantities

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

Nonconventional limits of random sequences related to partitions of integers

We deal with a sequence of integer-valued random variables $\{Z_N\}_{N=1}^{\infty}$ which is related to restricted partitions of positive integers. We observe that $Z_N=X_1+ \ldots + X_N$ for independent and bounded random variables $X_j$'s, so $Z_N$ has finite mean ${\bf E}Z_N$ and variance ${\bf Var}Z_N$. We want to find the limit distribution of ${\hat Z}_N=\left(Z_N-{\bf E}Z_N\right)/{\sqrt{{\bf Var}Z_N}}$ as $N \to \infty.$ While in many cases the limit distribution is normal, the main results established in this paper are that ${\hat Z}_N \overset{d}{\to} Z_{*},$ where $Z_{*}$ is a bounded random variable. We find explicitly the range of values of $Z_*$ and derive some properties of its distribution. The main tools used are moment generating functions, cumulant generating functions, moments and cumulants of the random variables involved. Useful related topics are also discussed.

Rounding of continuous random variables and oscillatory asymptotics

We study the characteristic function and moments of the integer-valued random variable ⌊X+α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integer-valued random variables. This leads to oscillatory terms in asymptotics for these variables, something that has often been observed, for example in the analysis of several algorithms. We give some examples, including applications to tries, digital search trees and Patricia tries.

A bound for the distribution of the sum of discrete associated or negatively associated random variables

Let $X_1, X_2,\dots, X_n$ be a sequence of integer-valued random variables that are either associated or negatively associated.We present a simple upper bound for the distance between the distribution of the sumof $X_i$ and a sum of $n$ independent randomvariables with the same marginals as $X_i$. An upper bound useful for establishing a compound Poisson approximation for $\Sigma_{i=1}^nX_i$ is also provided. The new bounds are of the same order as the much acclaimed Stein–Chen bound.

Existence and positivity of the limit in processes with a branching structure

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

Existence and positivity of the limit in processes with a branching structure

We define a stochastic process {Xn} based on partial sums of a sequence of integer-valued random variables (K0,K1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K1,K2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {cn} with cn+1/cn → a as n → ∞ such that Xn/cn converges almost surely to a finite random variable which is positive on the event {Xn ↛ 0}. The result is extended to the case of exchangeable summands.

Existence and positivity of the limit in processes with a branching structure

We define a stochastic process {X n } based on partial sums of a sequence of integer-valued random variables (K 0,K 1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K 1,K 2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {c n } with c n+1/c n → a as n → ∞ such that X n /c n converges almost surely to a finite random variable which is positive on the event {X n ↛ 0}. The result is extended to the case of exchangeable summands.

Asymptotic normality of processes with a branching structure

We define a stochastic process in terms of cumulative sums of the sequence of integer-valued random variables in such a way that has a branching structure; in particular when are iid and non-negativeis is a Bienayme-Galton-Watson branching process. We establish distributional limits involving the standard normal distribution for Xn and for , with a variety of standardizations and with various conditioning events. The interplay between the asymptotic behaviour of Xn and that of Tn is central to our investigations

Asymptotic normality of processes with a branching structure

We define a stochastic process in terms of cumulative sums of the sequence of integer-valued random variables in such a way that has a branching structure; in particular when are iid and non-negativeis is a Bienaymé-Galton-Watson branching process. We establish distributional limits involving the standard normal distribution for Xn and for , with a variety of standardizations and with various conditioning events. The interplay between the asymptotic behaviour of Xn and that of Tn is central to our investigations

Likelihood ratio and a class of strong laws for the sequence of integer-valued random variables

In this paper, the notion of likelihood ratio, as a measure of the deviation between a sequence of integer-valued random variables and an independent random sequence with Poisson distribution is introduced, and a class of strong laws, expressed by inequalities, on certain sets determined by this notion, are obtained. A class of strong laws for a sequence of independent Poisson random variables are special cases of the results of this paper.

Asymptotic normality for random sums of linear processes

Let X j = Σ ∞ j=0 b j ε i− j , for i= 1,2,3,…, be the consecutive obeservations of stationary linear processes, with Eε i =0, Eε 2 i = σ 2 and E| ε i | 3 < ∞, and let {N n,n∈ N} be a sequence of integer-valued random variables defined on the same probability space as {ε j, i∈ Z} . Under mild conditions on the process, asymptotic normality is established for the random sum, S N n= Σ N n i= 1X i , of linear processes. Statistical applications of the result are also discussed.

On estimation of generalized densities

This paper is concerned with the problem of estimating a generalized density of type (δ denotes the Dirac delta) corresponding to a probability distribution function without a singular part. We propose two natural classes of estimates and obtain consistency results with respect to suitable norms. Our approach could be useful to provide an alternative model for those problems (simulation, smoothed bootstrap) in which density estimates are used as auxiliary tools to draw artificial samples. In particular, we suggest the possibility of considering a sort of mixed bootstrap, placed in an intermediate position between the two usual versions (smoothed and unsmoothed) of this technique. Certain interesting sequences of integer-valued random variables [called rare numbers by E. Key (1991)] arise, in a natural way, in the consistency proofs.

FIRST‐ORDER INTEGER‐VALUED AUTOREGRESSIVE (INAR(1)) PROCESS

Abstract. A simple model for a stationary sequence of integer‐valued random variables with lag‐one dependence is given and is referred to as the integer‐valued autoregressive of order one (INAR(1)) process. The model is suitable for counting processes in which an element of the process at time t can be either the survival of an element of the process at time t ‐ 1 or the outcome of an innovation process. The correlation structure and the distributional properties of the INAR(1) model are similar to those of the continuous‐valued AR(1) process. Several methods for estimating the parameters of the model are discussed, and the results of a simulation study for these estimation methods are presented.

On sequential density estimation

We consider the problem of sequential estimation of a density function f at a point x0 which may be known or unknown. Let Tn be a sequence of estimators of x 0 . For two classes of density estimators f n , namely the kernel estimates and a recursive modification of these, we show that if N(d) is a sequence of integer-valued random variables and n(d) a sequence of constants with N(d)/n(d)→ 1 in probability as d → 0, then f N(d) (T N(d) -f(x0) is asymptotically normally distributed (when properly normed). We also propose two new classes of stopping rules based on the ideas of fixed-width interval estimation and show that for these rules, N(d)/n(d) → 1 almost surely and EN(d)/n(d) → 1 as d → 0. One of the stopping rules is itself asymptotically normally distributed when properly normed and yields a confidence interval for f(x0) of fixed-width and prescribed coverage probability.

On a Local Limit Theorem for Lattice Random Variables

Yu. V. Prokhorov made a conjecture that the two statements are equivalent: (a) the local limit theorem is applicable to a sequence of integer-valued random variables, (b) these variables obey the central limit theorem and their sums are asymptotically uniformly distributed modulo h for arbitrary h.We give an example showing that the above conjecture does not hold true.