Application of Smirnov Words to Waiting Time Distributions of Runs

In this paper, joint distributions of number of success runs of length k and number of failure runs of length k' are obtained by using combinatorial techniques including lattice path approach under Pólya-Eggenberger model. Some of its particular cases, for different values of the parameters, are derived. Sooner and later waiting time problems and joint distributions of number of success runs of various types until first occurrence of consecutive success runs of specified length under the model are obtained. The sooner and later waiting time problems for Bernoulli trials (see Ebneshahrashoob and Sobel [3]) and joint distributions of the type discussed by Uchiada and Aki [11] are shown as particular cases. Assuming Ln and Sn to be the lengths of longest and smallest success runs, respectively, in a sample of size n drawn by Pólya-Eggenberger sampling scheme, the joint distributions of Ln and Sn as well as distribution of M=max(Ln,Fn)n, where Fn is the length of longest failure run, are also obtained.

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The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach

Journal of Applied Probability ◽

10.1239/jap/1067436088 ◽

2003 ◽

Vol 40 (4) ◽

pp. 881-892 ◽

Cited By ~ 16

Author(s):

Valeri T. Stefanov

Keyword(s):

Markov Processes ◽

Waiting Time ◽

Generating Functions ◽

Continuous Time ◽

Finite Alphabet ◽

Algorithmic Approach ◽

First Occurrence ◽

Semi Markov Processes ◽

Pattern Occurrences ◽

Continuous Time Models

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

Download Full-text

The intersite distances between pattern occurrences in strings generated by general discrete- and continuous-time models: an algorithmic approach

Journal of Applied Probability ◽

10.1017/s0021900200020179 ◽

2003 ◽

Vol 40 (04) ◽

pp. 881-892 ◽

Cited By ~ 2

Author(s):

Valeri T. Stefanov

Keyword(s):

Markov Processes ◽

Waiting Time ◽

Generating Functions ◽

Continuous Time ◽

Finite Alphabet ◽

Algorithmic Approach ◽

First Occurrence ◽

Semi Markov Processes ◽

Pattern Occurrences ◽

Continuous Time Models

The formation of patterns from letters of a finite alphabet is considered. The strings of letters are generated by general discrete- and continuous-time models which embrace as particular cases all models considered in the literature. The letters of the alphabet are identified by the states of either discrete- or continuous-time semi-Markov processes. A new and unifying method is introduced for evaluation of the generating functions of both the intersite distance between occurrences of an arbitrary, but fixed, pattern and the waiting time until the first occurrence of that pattern. Our method also covers in a unified way relevant and important joint generating functions. Furthermore, our results lead to an easy and efficient implementation of the relevant evaluations.

Download Full-text