On the Poisson approximation of random diagonal sums of Bernoulli matrices

• Autorzy: Roos Bero

Identyfikatory

DOI

10.37190/0208-4147.00208

• Języki publikacji: EN

Abstrakty

EN

We use the Stein-Chen method to prove new explicit inequalities for the total variation, Wasserstein and local distances between the distribution of a random diagonal sum of a Bernoulli matrix and a Poisson distribution. Approximation results using a finite signed measure of higher order are given as well. Some of our total variation bounds improve existing results in the literature.

Słowa kluczowe

EN

approximation error; Hoeffding statistic; Poisson approximation

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2024
• Tom: Vol. 44, Fasc. 2
• Strony: 237--265
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Roos Bero

• FB IV–Mathematics, Trier University, 54286 Trier, Germany

Bibliografia

• [1] R. Adamczak, B. Polaczyk and M. Strzelecki, Modified log-Sobolev inequalities, Beckner inequalities and moment estimates, J. Funct. Anal. 282 (2022), art. 109349, 76 pp.
• [2] A. D. Barbour and G. K. Eagleson, Poisson approximation for some statistics based on ex-changeable trials, Adv. Appl. Probab. 15 (1983), 585-600.
• [3] A. D. Barbour and P. Hall, On the rate of Poisson convergence, Math. Proc. Cambridge Philos. Soc. 95 (1984), 473-480.
• [4] A. D. Barbour and L. Holst, Some applications of the Stein-Chen method for proving Poisson convergence, Adv. Appl. Probab. 21 (1989), 74-90.
• [5] A. D. Barbour, L. Holst and S. Janson, Poisson Approximation, Clarendon Press, Oxford, 1992.
• [6] A. D. Barbour and J. L. Jensen, Local and tail approximations near the Poisson limit, Scand.J. Statist. 16 (1989), 75-87.
• [7] A. D. Barbour, A. Röllin and N. Ross, Error bounds in local limit theorems using Stein’s method, Bernoulli 25 (2019), 1076-1104.
• [8] P. Brändén, J. Haglund, M. Visontai and D. G. Wagner, Proof of the monotone column permanent conjecture, in: Notions of Positivity and the Geometry of Polynomials, Springer, Basel, 2011, 63-78.
• [9] L. H. Y. Chen, Poisson approximation for dependent trials, Ann. Probab. 3 (1975), 534-545.
• [10] L. H. Y. Chen, An approximation theorem for sums of certain randomly selected indicators, Z. Wahrsch. Verw. Gebiete 33 (1975/76), 69-74.
• [11] P. Deheuvels and D. Pfeifer, A semigroup approach to Poisson approximation, Ann. Probab.14 (1986), 663-676.
• [12] P. Deheuvels and D. Pfeifer, On a relationship between Uspensky’s theorem and Poisson approximations, Ann. Inst. Statist. Math., 40 (1988), 671-681.
• [13] J. Kerstan, Verallgemeinerung eines Satzes von Prochorow und Le Cam, Z. Wahrsch. Verw. Gebiete 2 (1964), 173-179.
• [14] J. Pitman, Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Combin. Theory Ser. A 77 (1997), 279-303.
• [15] Yu. V. Prokhorov, Asymptotic behavior of the binomial distribution, Uspekhi Mat. Nauk (N.S.) 8 (1953), 135-142 (in Russian); English transl.: Selected Translations in Mathematical Statistics and Probability 1, Amer. Math. Soc., Providence, RI, 1961, 87-95.
• [16] B. Roos, Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution, Bernoulli 5 (1999), 1021-1034.
• [17] B. Roos, Binomial approximation to the Poisson binomial distribution: the Krawtchouk expansion, Teor. Veroyatnost. i Primenen. 45 (2000), 328-344; also Theory Probab. Appl. 45 (2001), 258-272.
• [18] B. Roos, Sharp constants in the Poisson approximation, Statist. Probab. Lett. 52 (2001), 155-168.
• [19] B. Roos, Improvements in the Poisson approximation of mixed Poisson distributions, J. Statist. Plann. Inference 113 (2003), 467-483.
• [20] B. Roos, Smoothness and Lévy concentration function inequalities for distributions of random diagonal sums, Theory Probab. Math. Statist. 111 (2024), 137-151.
• [21] S. Y. Shorgin, Approximation of a generalized binomial distribution, Teor. Veroyatnost. I Primenen. 22 (1977), 867-871 (in Russian); English transl.: Theory Probab. Appl. 22 (1977), 846-850.
• [22] C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in: Proc. 6th Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1970/1971), Vol. II: Probability Theory, Univ. of California Press, Berkeley, CA, 1972, 583-602