Some Elementary Results on Poisson Approximation in a Sequence of Bernoulli Trials

SIAM Review ◽  
1978 ◽  
Vol 20 (3) ◽  
pp. 567-579 ◽  
Author(s):  
R. J. Serfling
Keyword(s):  
Poisson Approximation ◽  
Bernoulli Trials

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

Poisson approximations for runs and patterns of rare events

1991 ◽  
Vol 23 (4) ◽  
pp. 851-865 ◽  
Author(s):  
Anant P. Godbole
Keyword(s):  
Success Probability ◽  
Standard Condition ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Poisson Random Variable ◽  
Runs And Patterns ◽  
Reliability Systems ◽  
Poisson Approximations

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk→λ. This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk–m) for some m = 0, 2, ···, k – 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.

Poisson approximations for runs and patterns of rare events

1991 ◽  
Vol 23 (04) ◽  
pp. 851-865 ◽  
Author(s):  
Anant P. Godbole
Keyword(s):  
Success Probability ◽  
Standard Condition ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Poisson Random Variable ◽  
Runs And Patterns ◽  
Reliability Systems ◽  
Poisson Approximations

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk →λ. This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk–m ) for some m = 0, 2, ···, k – 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.

Asymptotic results for the multiple scan statistic

2017 ◽  
Vol 54 (1) ◽  
pp. 320-330 ◽  
Author(s):  
M. V. Boutsikas ◽  
M. V. Koutras ◽  
F. S. Milienos
Keyword(s):  
Random Walk ◽  
Real Life ◽  
Poisson Approximation ◽  
Scan Statistics ◽  
Bernoulli Trials ◽  
Scan Statistic ◽  
Asymptotic Results ◽  
Compound Poisson Approximation ◽  
Multiple Scan ◽  
Random Walk Theory

AbstractThe contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.

Poisson approximation for (k 1, k 2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (04) ◽  
pp. 1081-1092
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k 1, k 2) denote the number of times that k 1 failures are followed by k 2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k 1, k 2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for N k 3 (n; k 1, k 2), the number of times that k 1 failures followed by k 2 successes occur k 3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

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