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# Poisson approximation for random sums of Bernoulli random variables

Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden
(Impact Factor: 0.53). 02/1991; DOI: 10.1016/0167-7152(91)90135-E

ABSTRACT Bounds for the total variation distance between the distribution of the sum of a random number of Bernoulli summands and an appropriate Poisson distribution are given. The results can be used to derive limit theorems with rates of convergence for marked and thinned point processes. Some examples are given.

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##### Article: BINOMIAL APPROXIMATION FOR RANDOM SUMS OF BERNOULLI RANDOM VARIABLES
08/2014; 94(5). DOI:10.12732/ijpam.v94i5.7
• ##### Article: Poisson approximation for (k1, k2)events via the Stein-Chen method
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ABSTRACT: 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.
Journal of Applied Probability 12/2004; 41(2004). DOI:10.1239/jap/1101840553 · 0.69 Impact Factor
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##### Article: AN IMPROVED POISSON APPROXIMATION FOR BERNOULLI RANDOM SUMMANDS
08/2014; 94(5). DOI:10.12732/ijpam.v94i5.6