Poisson approximation for (k 1, k 2)-events via the Stein-Chen method
| Autor: | Palaniappan Vellaisamy |
|---|---|
| Rok vydání: | 2004 |
| Předmět: |
Statistics and Probability
Limit Theorem Binomial Distribution Of Order (K(1) K(2)) General Mathematics Runs Poisson distribution 01 natural sciences Upper and lower bounds Combinatorics 010104 statistics & probability symbols.namesake Probability theory Order-K Bernoulli trial (K1 K2)-Event Limit (mathematics) 0101 mathematics Patterns Mathematics 010102 general mathematics Stein-Chen Method Random-Variables Rate of convergence Markov-Bernoulli Variable symbols Probability distribution Distributions Poisson Approximation Rate Of Convergence Bernoulli process Statistics Probability and Uncertainty |
| Zdroj: | Journal of Applied Probability. 41:1081-1092 |
| ISSN: | 1475-6072 0021-9002 |
| DOI: | 10.1017/s0021900200020842 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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