Compound Poisson Approximations to Sums of Extrema of Bernoulli Variables
Tóm tắt
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.
Tài liệu tham khảo
A.D. Barbour, L.H.Y. Chen, and W.-L. Loh, Compound Poisson approximation for nonnegative random variables via Stein’s method, Ann. Probab., 20(4):1843–1866, 1992.
A.D. Barbour and A. Xia, Poisson perturbations, ESAIM, Probab. Stat., 3:131–150, 1999.
V. Čekanavičius, Approximation Methods in Probability Theory, Universitext, Springer, Cham, 2016.
V. Čekanavičius and P. Vellaisamy, Discrete approximations for sums ofm-dependent random variables, ALEA, Lat. Am. J. Probab. Math. Stat., 12(2):765–792, 2015.
V. Čekanavičius and P. Vellaisamy, Lower bounds for discrete approximations to sums of m-dependent random variables, Probab. Math. Stat., 40(1):23–35, 2020.
F. Daly, Compound Poisson approximation with association or negative association via Stein’s method, Electron. Commun. Probab., 18:30, 2013.
H.L. Gan and A. Xia, Stein’s method for conditional compound Poisson approximation, Stat. Probab. Lett., 100:19–26, 2015.
L. Heinrich, Factorization of the characteristic function of a sum of dependent random variables, Lith. Math. J., 22(1):92–100, 1982.
A.N. Kumar, N.S. Upadhye, and P. Vellaisamy, Approximations related to the sums of m-dependent random variables, Braz. J. Probab. Stat., 36(2):349–368, 2022.
J. Petrauskienė and V. Čekanavičius, Compound Poisson approximations for sums of 1-dependent random variables. I, Lith. Math. J., 50(3):323–336, 2010.
N. Ross, Fundamentals of Stein’s method, Probab. Surv., 8:210–293, 2011.
A. Röllin, Approximation of sums of conditionally independent variables by the translated Poisson distribution, Bernoulli, 11(6):1115–1128, 2005.
N.S. Upadhye and A.N. Kumar, Pseudo-binomial approximation to (k1, k2)-runs, Stat. Probab. Lett., 141:19–30, 2018.
P. Vellaisamy, Poisson approximation for (k1, k2) events via the Stein–Chen method, J. Appl. Probab., 41(4):1081–1092, 2004.
X. Wang and A. Xia, On negative binomial approximation to k-runs, J. Appl. Probab., 45(2):456–471, 2008.