On the distribution of the longest success-run in Bernoulli trials

2016 ◽  
Vol 20 (34) ◽  
Author(s):  
Bolesław Kopocinski
Keyword(s):  
Bernoulli Trials ◽  
Success Run ◽  
Longest Success Run

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.

Large deviations for longest runs in Markov chains

2020 ◽  
Vol 26 (2) ◽  
pp. 309-314
Author(s):  
Zhenxia Liu ◽  
Yurong Zhu
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
Markov Chains ◽  
Present Note ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Large Deviation Principles ◽  
Success Run ◽  
Longest Success Run ◽  
Math Statist

AbstractWe continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].

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”.

On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

2003 ◽  
Vol 40 (02) ◽  
pp. 346-360 ◽  
Author(s):  
James C. Fu ◽  
Liqun Wang ◽  
W. Y. Wendy Lou
Keyword(s):  
Large Deviation ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Longest Run ◽  
Exact Distributions ◽  
Markov Dependent Trials ◽  
Success Run ◽  
Longest Success Run ◽  
Theoretical Results

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

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.

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 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.

On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

2003 ◽  
Vol 40 (2) ◽  
pp. 346-360 ◽  
Author(s):  
James C. Fu ◽  
Liqun Wang ◽  
W. Y. Wendy Lou
Keyword(s):  
Large Deviation ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Longest Run ◽  
Exact Distributions ◽  
Markov Dependent Trials ◽  
Success Run ◽  
Longest Success Run ◽  
Theoretical Results

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

scholarly journals On the longest runs in Markov chains

2018 ◽  
Vol 38 (2) ◽  
pp. 407-428 ◽  
Author(s):  
Zhenxia Liu ◽  
Xiangfeng Yang
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Generating Function ◽  
Cumulative Distribution Function ◽  
Moment Generating Function ◽  
Cumulative Distribution ◽  
Global Estimate ◽  
Success Run ◽  
Longest Success Run ◽  
The Moment

In the first n steps of a two-state success and failure Markov chain, the longest success run Ln has been attracting considerable attention due to its various applications. In this paper, we study Ln in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of Ln proposed in this paper, and the proofs of the large deviations include the Gärtner–Ellis theorem and the moment generating function.

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