Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials

In this paper we study exact distributions of sooner and later waiting times for runs in Markov dependent bivariate trials. We give systems of linear equations with respect to conditional probability generating functions of the waiting times. By considering bivariate trials, we can treat very general and practical waiting time problems for runs of two events which are not necessarily mutually exclusive. Numerical examples are also given in order to illustrate the feasibility of our results.     

Joint Distributions of Runs in a Sequence of Multi-State Trials

In this paper we introduce a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variable of returnable type and discuss some of their properties. These concepts are extensions of the Markov chain imbeddable random variable of binomial type which was introduced and developed by Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743–766). By using the results, we obtain the distributions and the probability generating functions of numbers of occurrences of runs of a specified length based on four different ways of counting in a sequence of multi-state trials. Our results also yield the distribution of the waiting time problems.     

Sooner and Later Waiting Time Problems Based on a Dependent Sequence

This paper introduces a new concept: a binary sequence of order (k,r), which is an extension of a binary sequence of order k and a Markov dependent sequence. The probability functions of the sooner and later waiting time random variables are derived in the binary sequence of order (k,r). The probability generating functions of the sooner and later waiting time distributions are also obtained. Extensions of these results to binary sequence of order (g,h) are also presented.     

Joint distributions of runs in a sequence of higher-order two-state Markov trials

Consider a time homogeneous {0, 1}-valued m-dependent Markov chain . In this paper, we study the joint probability distribution of number of 0-runs of length and number of 1-runs of length in n trials. We study the joint distributions based on five popular counting schemes of runs. The main tool used to obtain the probability generating function of the joint distribution is the conditional probability generating function method. Further a compact method for the evaluation of exact joint distribution is developed. For higher-order two-state Markov chain, these joint distributions are new in the literature of distributions of run statistics. We use these distributions to derive some waiting time distributions.     

Joint distributions of numbers of success-runs and failures until the first consecutivek successes

Joint distributions of the numbers of failures, successes and success-runs of length less thank until the first consecutivek successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of orderk. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.This research was partially supported by the ISM Cooperative Research Program (93-ISM-CRP-8).     

Generalized binomial and negative binomial distributions of order<Emphasis Type="Italic">k</Emphasis> by the<Emphasis Type="Italic">l</Emphasis>-overlapping enumeration scheme

In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete distributions.     

Joint distributions associated with patterns,successes and failures in a sequence of multi-state trials

Let {Z _t ,t≥1} be a sequence of trials taking values in a given setA={0, 1, 2,...,m}, where we regard the value 0 as failure and the remainingm values as successes. Let ε be a (single or compound) pattern. In this paper, we provide a unified approach for the study of two joint distributions, i.e., the joint distribution of the numberX _n of occurrences of ε, the numbers of successes and failures inn trials and the joint distribution of the waiting timeT _r until ther-th occurrence of ε, the numbers of successes and failures appeared at that time. We also investigate some distributions as by-products of the two joint distributions. Our methodology is based on two types of the random variablesX _n (a Markov chain imbeddable variable of binomial type and a Markov chain imbeddable variable of returnable type). The present work develops several variations of the Markov chain imbedding method and enables us to deal with the variety of applications in different fields. Finally, we discuss several practical examples of our results. This research was partially supported by the ISM Cooperative Research Program (2002-ISM·CRP-2007).     

Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials

Let Z ₀, Z ₁,...,Z _n be a sequence of Markov dependent trials with state space Ω = {F ₁,...,F _λ, S ₁,...,S _ν}, where we regard F ₁,...,F _λ as failures and S ₁,...,S _ν as successes. In this paper, we study the joint distribution of the numbers of S _i -runs of lengths k _ij (i = 1,2,...,ν, j = 1,2,...,r _i ) based on four different enumeration schemes. We present formulae for the evaluation of the probability generating functions and the higher order moments of this distribution. In addition, when the underlying sequence is i.i.d. trials, the conditional distribution of the same run statistics, given the numbers of success and failure is investigated. We give further insights into the multivariate run-related problems arising from a sequence of the multistate trials. Besides, our results have potential applications to problems of various research areas and will come to prominence in the future. This research was partially supported by the ISM Cooperative Research Program (2004-ISM·CRP-2007).     

Generalized Waiting Time Problems Associated with Pattern in Polya's Urn Scheme

Let X ₁, X ₂, ... be a sequence obtained by Polya's urn scheme. We consider a waiting time problem for the first occurrence of a pattern in the sequence X ₁, X ₂, ... , which is generalized by a notion score. The main part of our results is derived by the method of generalized probability generating functions. In Polya's urn scheme, the system of equations is composed of the infinite conditional probability generating functions, which can not be solved. Then, we present a new methodology to obtain the truncated probability generating function in a series up to an arbitrary order from the system of infinite equations. Numerical examples are also given in order to illustrate the feasibility of our results. Our results in this paper are not only new but also a first attempt to treat the system of infinite equations.     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

On Number of Occurrences of Success Runs of Specified Length in a Higher-Order Two-State Markov Chain

Let X^-m+1, X^-m+2,..., X⁰, X¹, X²,..., Xⁿ be a time-homogeneous {0, 1}-valued m-th order Markov chain. The probability distributions of numbers of runs of "1" of length k (k m) and of "1" of length k (k < m) in the sequence of a {0, 1}-valued m-th order Markov chain are studied. There are some ways of counting numbers of runs with length k. This paper studies the distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length k, the number of runs with length greater than or equal to k, the number of overlapping runs of length k and the number of runs of length exactly k.     

Derivation of the probability distribution functions for succession quota random variables

The probability distribution functions (pdf's) of the sooner and later waiting time random variables (rv's) for the succession quota problem (k successes and r failures) are derived presently in the case of a binary sequence of order k. The probability generating functions (pgf's) of the above rv's are then obtained directly from their pdf's. In the case of independent Bernoulli trials, expressions for the pdf's in terms of binomial coefficients are also established.     

Joint Distributions of Numbers of Success-runs Until the First Consecutive k Successes in a Higher-Order Two-State Markov Chain

Let X-m+1, X-m+2,.., X0, X1, X2,.., be a time-homogeneous {0, 1}-valued m-th order Markov chain. Joint distributions of the numbers of trials, failures and successes, of the numbers of trials and success-runs of length l (m l k) and of the numbers of trials and success-runs of length l (l m k) until the first consecutive k successes are obtained in the sequence X1, X2,.., There are some ways of counting numbers of runs of length l. This paper studies the joint distributions based on four ways of counting numbers of runs, i.e., the number of non-overlapping runs of length l, the number of runs of length greater than or equal to l, the number of overlapping runs of length l and the number of runs of length exactly l. Marginal distributions of them can be obtained immediately, and surprisingly their distributions are very simple.     

A generalized Pólya urn model and related multivariate distributions

In this paper, we study of Pólya urn model containing balls of (m+1) different labels under a general replacement scheme, which is characterized by an (m+1) × (m+1) addition matrix of integers without constraints on the values of these (m+1)² integers other than non-negativity. LetX ₁,X ₂,...,X _n be trials obtained by the Pólya urn scheme (with possible outcomes: “O”, “1”,...“m”). We consider the multivariate distributions of the numbers of occurrences of runs of different types arising from the various enumeration schemes and give a recursive formula of the probability generating function. Some closed form expressions are derived as special cases, which have potential applications to various areas. Our methods for the derivation of the multivariate run-related distribution are very simple and suitable for numerical and symbolic calculations by means of computer algebra systems. The results presented here develop a general workable framework for the study of Pólya urn models. Our attempts are very useful for understanding non-classic urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results. This research was partially supported by the ISM Cooperative Research Program (2003-ISM·CRP-2007).     

Probability Distribution Functions of Succession Quotas in the Case of Markov Dependent Trials

The exact probability distribution functions (pdf's) of the sooner andlater waiting time random variables (rv's) for the succession quota problemare derived presently in the case of Markov dependent trials. This is doneby means of combinatorial arguments. The probability generating functions(pgf's) of these rv's are then obtained by means of enumerating generatingfunctions (enumerators). Obvious modifications of the proofs provideanalogous results for the occurrence of frequency quotas and such a resultis established regarding the pdf of a frequency and succession quotas rv.Longest success and failure runs are also considered and their jointcumulative distribution function (cdf) is obtained.     

On waiting time distributions associated with compound patterns in a sequence of multi-state trials

In this article, waiting time distributions of compound patterns are considered in terms of the generating function of the numbers of occurrences of the compound patterns. Formulae for the evaluation of the generating functions of waiting time are given, which are very effective computational tools. We provide several viewpoints on waiting time problems associated with compound patterns and develop a general workable framework for the study of the corresponding distributions. The general theory is employed for the investigation of some examples in order to illustrate how the distributions of waiting time can be derived through our theoretical results. This research was partially supported by the ISM Cooperative Research Program (2006-ISM·CRP-2007).     

The Exact and Limiting Distributions for the Number of Successes in Success Runs Within a Sequence of Markov-Dependent Two-State Trials

The total number of successes in success runs of length greater than or equal to k in a sequence of n two-state trials is a statistic that has been broadly used in statistics and probability. For Bernoulli trials with k equal to one, this statistic has been shown to have binomial and normal distributions as exact and limiting distributions, respectively. For the case of Markov-dependent two-state trials with k greater than one, its exact and limiting distributions have never been considered in the literature. In this article, the finite Markov chain imbedding technique and the invariance principle are used to obtain, in general, the exact and limiting distributions of this statistic under Markov dependence, respectively. Numerical examples are given to illustrate the theoretical results.     

Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials

Let {Z _n , n 1} be a time-homogeneous {0, 1}-valued Markov chain, and let N _n be a random variable denoting the number of runs of "1" of length k in the first n trials. In this article we conduct a systematic study of N _n by establishing formulae for the evaluation of its probability generating function, probability mass function and moments. This is done in three different enumeration schemes for counting runs of length k, the "non-overlapping", the "overlapping" and the "at least" scheme. In the special case of i.i.d. trials several new results are established.