Success runs in a two-state Markov chain

1974 ◽  
Vol 11 (1) ◽  
pp. 190-192 ◽  
Author(s):  
M. B. Rajarshi
Keyword(s):  
Markov Chain ◽  
Generating Function ◽  
Waiting Time ◽  
Recurrent Events ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Fixed Length ◽  
First Occurrence ◽  
Success Run

Success runs of a fixed length in a two-state Markov chain are discussed. The results are analogous to those in the case of independent Bernoulli trials. The generating function of the waiting time for the first occurrence of a success run is obtained from the theory of recurrent events. Under certain conditions, the distribution of the number of long runs in a large number of trials is shown to be Poisson.

Success runs in a two-state Markov chain

1974 ◽  
Vol 11 (01) ◽  
pp. 190-192 ◽  
Author(s):  
M. B. Rajarshi
Keyword(s):  
Markov Chain ◽  
Generating Function ◽  
Waiting Time ◽  
Recurrent Events ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Fixed Length ◽  
First Occurrence ◽  
Success Run

Success runs of a fixed length in a two-state Markov chain are discussed. The results are analogous to those in the case of independent Bernoulli trials. The generating function of the waiting time for the first occurrence of a success run is obtained from the theory of recurrent events. Under certain conditions, the distribution of the number of long runs in a large number of trials is shown to be Poisson.

Lengths of runs and waiting time distributions by using Pólya-Eggenberger sampling scheme

2002 ◽  
Vol 39 (3-4) ◽  
pp. 309-332 ◽  
Author(s):  
K. Sen ◽  
Manju L. Agarwal ◽  
S. Chakraborty
Keyword(s):  
Waiting Time ◽  
Lattice Path ◽  
Sampling Scheme ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Waiting Time Distributions ◽  
Joint Distributions ◽  
First Occurrence

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.

Recurrent composite events

1967 ◽  
Vol 4 (01) ◽  
pp. 34-61 ◽  
Author(s):  
R.T. Leslie
Keyword(s):  
Recurrent Events ◽  
Recurrent Event ◽  
Complex Pattern ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Trial Number ◽  
Composite Events ◽  
Success Run ◽  
Definition Of ◽  
Two Alternatives

On a sequence of Bernoulli trials, the definition of a recurrent event ε involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial. Success runs are the best known of such recurrent events, but Feller (1959, §13.8) mentions more complicated patterns, among which two types may be distinguished. The simpler involves a single more complex pattern such as SSFFSS; the second type involves a set of alternative events defining ε, which is said to occur when any one of the alternatives occurs at trial number n. Thus if ε stands for “either a success run of length r or a failure run of length ρ”, there are two alternatives in the set; the problem is elementary because the component events are “non-overlapping”.

Recurrent composite events

1967 ◽  
Vol 4 (1) ◽  
pp. 34-61 ◽  
Author(s):  
R.T. Leslie
Keyword(s):  
Recurrent Events ◽  
Recurrent Event ◽  
Complex Pattern ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Trial Number ◽  
Composite Events ◽  
Success Run ◽  
Definition Of ◽  
Two Alternatives

On a sequence of Bernoulli trials, the definition of a recurrent event ε involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial. Success runs are the best known of such recurrent events, but Feller (1959, §13.8) mentions more complicated patterns, among which two types may be distinguished. The simpler involves a single more complex pattern such as SSFFSS; the second type involves a set of alternative events defining ε, which is said to occur when any one of the alternatives occurs at trial number n. Thus if ε stands for “either a success run of length r or a failure run of length ρ”, there are two alternatives in the set; the problem is elementary because the component events are “non-overlapping”.

scholarly journals Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains

2006 ◽  
Vol 43 (01) ◽  
pp. 127-140 ◽  
Author(s):  
Joseph Glaz ◽  
Martin Kulldorff ◽  
Vladimir Pozdnyakov ◽  
J. Michael Steele
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Function ◽  
Waiting Time ◽  
Waiting Times ◽  
Higher Order ◽  
Expected Value ◽  
Finite Collection

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

scholarly journals Success Run Waiting Times and Fuss-Catalan Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
S. J. Dilworth ◽  
S. R. Mane
Keyword(s):  
Power Series ◽  
Waiting Time ◽  
Geometric Distribution ◽  
Waiting Times ◽  
Exact Expression ◽  
Catalan Numbers ◽  
Auxiliary Equation ◽  
Bernoulli Trials ◽  
Asymptotic Results ◽  
Success Run

We present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run ofkconsecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of orderk. We show that the series coefficients are Fuss-Catalan numbers and write the roots in terms of the generating function of the Fuss-Catalan numbers. Our main result is a new exact expression for the distribution, which is more concise than previously published formulas. Our work extends the analysis by Feller, who gave asymptotic results. We obtain quantitative improvements of the error estimates obtained by Feller.

scholarly journals Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains

2006 ◽  
Vol 43 (1) ◽  
pp. 127-140 ◽  
Author(s):  
Joseph Glaz ◽  
Martin Kulldorff ◽  
Vladimir Pozdnyakov ◽  
J. Michael Steele
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generating Function ◽  
Waiting Time ◽  
Waiting Times ◽  
Higher Order ◽  
Expected Value ◽  
Finite Collection

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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