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Compound Poisson Approximations to Sums of Extrema of Bernoulli Variables

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Abstract

Let Sn = X1 + X2 + · · · + Xn, where Xj = max(ξj, ξj+1), and ξ1, ξ2, . . . , ξn+1 are independent Bernoulli random variables. If all P(ξj = 1) are small, then we approximate Sn by a compound Poisson random variable with two matching moments. If all P(ξj = 1) are large, then we apply compound Poisson and negative binomial approximations to n − Sn. We estimate the accuracy of approximation in the total-variation and Kolmogorov metrics. We also show that similar results hold for sums of minima of Bernoulli variables. In the proofs, we use Heinrich’s method.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania

    Gabija Liaudanskaite & Vydas Čekanavičius

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  1. Gabija Liaudanskaite
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  2. Vydas Čekanavičius
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Correspondence to Gabija Liaudanskaite.

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Liaudanskaite, G., Čekanavičius, V. Compound Poisson Approximations to Sums of Extrema of Bernoulli Variables. Lith Math J 62, 481–499 (2022). https://doi.org/10.1007/s10986-022-09571-y

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  • DOI: https://doi.org/10.1007/s10986-022-09571-y

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