Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials

Abstract

In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1)<i> i.e</i>. we have a sequence {(X_n/Y_n), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (<i>X</i>⁰_n,k¹₀,<i>X</i>¹_n,k¹₁;<i>Y</i>⁰_n,k²₀,<i>Y</i>¹_n,k²₁) where for <i>i</i>=0,1,<i>X</i>^<i>i</i>_{n,k¹_<i>i</i>} denotes the number of occurrences of <i>i</i>-runs of length <i>k</i>¹_<i>i</i> in the first component and <i>Y</i>^<i>i</i>_{n,k²_<i>i</i>} denotes the number of occurrences of <i>i</i>-runs of length <i>k</i>²_<i>i</i> in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ₁ and Λ₂ of lengths <i>k</i>₁ and <i>k</i>₂ respectively and obtain the pgf of joint distribution of (<i>X</i>_{<i>n</i>,Λ ₁},<i>Y</i>_{<i>n</i>,Λ₂} ) using method of conditional probability generating functions where <i>X</i>_{<i>n</i>,Λ₁}(<i>Y</i>_{<i>n</i>,Λ₂}) denotes the number of occurrences of pattern Λ₁(Λ₂ ) of length <i>k</i>₁ (<i>k</i>₂) in the first (second) <i>n</i> components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.