## Probability Surveys

### Poisson approximation

S. Y. Novak

#### Abstract

We overview results on the topic of Poisson approximation that are missed in existing surveys. The main attention is paid to the problem of Poisson approximation to the distribution of a sum of Bernoulli and, more generally, non-negative integer-valued random variables.

We do not restrict ourselves to a particular method, and overview the whole range of issues including the general limit theorem, estimates of the accuracy of approximation, asymptotic expansions, etc. Related results on the accuracy of compound Poisson approximation are presented as well.

We indicate a number of open problems and discuss directions of further research.

#### Article information

Source
Probab. Surveys, Volume 16 (2019), 228-276.

Dates
First available in Project Euclid: 14 August 2019

https://projecteuclid.org/euclid.ps/1565748164

Digital Object Identifier
doi:10.1214/18-PS318

Mathematical Reviews number (MathSciNet)
MR3992498

Zentralblatt MATH identifier
07104701

#### Citation

Novak, S. Y. Poisson approximation. Probab. Surveys 16 (2019), 228--276. doi:10.1214/18-PS318. https://projecteuclid.org/euclid.ps/1565748164

#### References

• [1] Adler R.J. (1978) Weak convergence results for extremal processes generated by dependent random variables. — Ann. Probab., v. 6, No 4, 660–667.
• [2] Arratia R., Goldstein L. and Gordon L. (1989) Two moments suffice for Poisson approximation. — Ann. Probab., v. 17, No 1, p. 9–25.
• [3] Arratia R., Gordon L. and Waterman M.S. (1990) The Erdös–Rényi law in distribution, for coin tossing and sequence matching. — Ann. Statist., v. 18, No 2, p. 539–570.
• [4] Arak T.V. (1981) On the convergence rate in Kolmogorov’s uniform limit theorem. — Theory Probab. Appl., v. 26, 219–239, 437–451.
• [5] Arak T.V. and Zaitsev A.Yu. (1984) Uniform limit theorems for sums of independent random variables. — Proc. Steklov Inst. Math., v. 174, 3–214.
• [6] Arenbaev N.K. (1976) Asymptotic behavior of the multinomial distribution. — Theory Probab. Appl., v. 21, 805–810.
• [7] Balakrishnan N. and Koutras M.V. (2001) Runs and scans with applications. New York: Wiley.
• [8] Banis R. (1985) A Poisson limit theorem for rare events of a discrete random field. — Litovsk. Mat. Sb., v. 25, No 1, 3–8.
• [9] Barbour A.D. and Eagleson G.K. (1983) Poisson approximation for some statistics based on exchangeable trials. — Adv. Appl. Probab., v. 15, No 3, 585–600.
• [10] Barbour A.D. (1987) Asymptotic expansions in the Poisson limit theorem. — Ann. Probab., v. 15, No 2, 748–766.
• [11] Barbour A.D. and Jensen J.L. (1989) Local and tail approximations near the Poisson limit. — Scand. J. Statist., v. 16, 75–87.
• [12] Barbour A.D., Holst L. and Janson S. (1992) Poisson Approximation. Oxford: Clarendon Press.
• [13] Barbour A.D., Chen L.H.Y. and Choi K.P. (1995) Poisson approximation for unbounded functions: independent summands. — Statistica Sinica, v. 5, No 2, 749–766.
• [14] Barbour A.D. and Utev S.A. (1999) Compound Poisson approximation in total variation. — Stochastic Process. Appl., v. 82, 89–125.
• [15] Barbour A.D. and Chryssaphinou O. (2001) — Ann. Appl. Probab., v. 11, No. 3, 964–1002.
• [16] Barbour A.D. and Čekanavičius V. (2002) Total variation asymptotics for sums of independent integer random variables. — Ann. Probab., v. 30, No 2, 509–545.
• [17] Barbour A.D., Novak S.Y. and Xia A. (2002) Compound Poisson approximation for the distribution of extremes. — Adv. Appl. Probab., v. 34, No 1, 223–240.
• [18] Barbour A.D. and Xia A. (1999) Poisson perturbations. — ESAIM Probab. Statist., v. 3, 131–150.
• [19] Barbour A.D. and Xia A. (2006) On Stein’s factors for Poisson approximation in Wasserstein distance. — Bernoulli, v. 6, 943–954.
• [20] Bernstein S.N. (1926) Sur l’extensiori du theoreme limite du calcul des probabilites aux sommes de quantites dependantes. — Math. Annalen, v. 97, 1–59.
• [21] Bernstein S.N. (1946) Probability Theory. Moscow: Nauka.
• [22] Borisov I.S. (1996) Poisson approximation of the partial sum process in Banach spaces. — Sibirsk. Math. J., v. 37, No 4, 627–634.
• [23] Borisov I.S. and Ruzankin P.S. (2002) Poisson approximation for expectations of unbounded functions of independent random variables. — Ann. Probab., v. 30, No 4, 1657–1680.
• [24] Borisov I.S. (2003) A remark on a theorem of Dobrushin and couplings in the Poisson approximation in Abelian groups. — Theory Probab. Appl., v. 48, No 3, 521–528.
• [25] Borisov I.S. and Vorozheikin I.S. (2008) Accuracy of approximation in the Poisson theorem in terms of $\chi ^{2}$ distance. — Sibir. Math. J., v. 49, No 1, 5–17.
• [26] Borovkov K. A. (1988) On the problem of improving Poisson approximation. — Theory Probab. Appl., v. 33, No 2, 343–347.
• [27] Brown T.C. (1983) Some Poisson approximations using compensators. — Ann. Probab. 11, 726–744.
• [28] Chen L.H.Y. (1975) Poisson approximation for dependent trials. — Ann. Probab., v. 3, 534–545.
• [29] Čekanavičius V. (1998) Poisson approximations for sequences of random variables. — Statist. Probab. Lett., v. 39, 101–107.
• [30] Čekanavičius V. and Kruopis J. (2000) Signed Poisson approximation: a possible alternative to normal and Poisson laws. — Bernoulli, v. 6, No 4, 591–606.
• [31] Čekanavičius V. and Vaitkus P. (2001) A centered Poisson approximation via Stein’s method. — Lithuanian Math. J., v. 41, No 4, 319–329.
• [32] Čekanavičius V. and Wang Y.H. (2003) Compound Poisson approximation for sums of discrete nonlattice random variables. — Adv. Appl. Probab., v. 35, 228–250.
• [33] Čekanavičius V. and Roos B. (2006) An expansion in the exponent for compound binomial approximations. — Liet. Matem. Rink., v. 46, 67–110.
• [34] Čekanavičius V. and Vellaisamy P. (2010) Compound Poisson and signed compound Poisson approximations to the Markov binomial law. — Bernoulli, v. 16, No 4, 1114–1136.
• [35] Chryssaphinou O., Papastavridis S. and Vaggelatou E. (2001) Poisson approximation for the nonoverlapping appearances of several words in Markov chains. — Combinator. Probab. Computing, v. 10, no. 4, 293–308.
• [36] Csaki E. and Földes A. (1996) On the length of the longest monotone block. — Studia Scientiarum Mathematicarum Hungarica, v. 31, 35–46.
• [37] Dabrowski A., Ivanoff G. and Kulik R. (2009) Some notes on Poisson limits for empirical point processes. — Canadian J. Statist., v. 37, 347–360.
• [38] Daley D.J. and Vere-Jones D. (2008) An introduction to the theory of point processes, v. II. General Theory and Structure. New York: Springer.
• [39] Dall’ Aglio G. (1956) Sugli estremi dei momenti delle funzioni di ripartizione dopia. — Ann. Scuola Normale Superiore Di Pisa, ser. Cl. Sci., v. 3, No 1, 33–74.
• [40] Deheuvels P. and Pfeifer D. (1986) A semigroup approach to Poisson approximation. — Ann. Probab., v. 14, No 2, 663–676.
• [41] Deheuvels P. and Pfeifer D. (1988) Poisson approximation of distributions and point processes. — J. Multivar. Anal., v. 25, 65–89.
• [42] Deheuvels P. and Pfeifer D. (1988) On a relationship between Uspensky’s theorem and Poisson approximation. — Ann. Inst. Statist. Math., v. 40, 671–681.
• [43] Deheuvels P., Pfeifer D. and Puri M.L. (1989) A new semigroup technique in Poisson approximation. — Semigroup Forum, v. 38, 189–201.
• [44] Denzel G.E. and O’Brien G.L. (1975) Limit theorems for extreme values of chain–dependent processes. — Ann. Probab., v. 3, No 5, 773–779.
• [45] Dobrushin R. L. (1970) Prescribing a system of random variables by conditional distributions. — Theory Probab. Appl., v. 15, No 3, 458–486.
• [46] Franken P. (1964) Approximation der verteilungen von summen unabhängiger nichtnegativer ganzzahliger zufallsgrössen durch Poissonsche verteilunged. — Mathematische Nachrichten, v. 27, 303–340.
• [47] Galambos J. (1987) The asymptotic theory of extreme order statistics. Melbourne: R.E.Krieger Publishing Co.
• [48] Gerber H.U. (1979) An introduction to mathematical risk theory. Philadelphia: Huebner Foundation.
• [49] Gini C. (1914) Di unà misura delle relazioni tra le graduatorie di due caratteri. — In: Appendix to Hancini A. Le Elezioni Generali Politiche del 1913 nel Comune di Roma. Rome: Ludovico Cecehini.
• [50] Gnedenko B.V. (1938) Uber die konvergenz der verteilungsgesetze von summen voneinander unabhangiger summanden. — C.R. Acad. Sci. URSS, v. 18, 231–234.
• [51] Gnedenko B.V. (1943) Sur la distribution du terme maximum d’une série aléatoire. — Ann. Math., v. 44, 423–453.
• [52] Gnedenko B.V. and Kolmogorov A.N. (1954) Limit distributions for sums of independent random variables. Addison-Wesley: Cambridge, MA.
• [53] Harremoës P. and Ruzankin P.S. (2004) Rate of convergence to Poisson law in terms of information divergence. — IEEE Trans. Inform Theory, v. 50, No 9, 2145–2149.
• [54] Haight F.A. (1967) Handbook of the Poisson distribution. New York: Wiley.
• [55] Herrmann H. (1965) Variationsabstand zwischen der Verteilung einer Summe unabhängiger nichtnegativer ganzzahliger Zufallsgrössen und Poissonschen Verteilungen. — Mathematische Nachrichten, v. 29, No 5, 265–289.
• [56] Hsing T., Hüsler J. and Leadbetter M.R. (1988) On the exceedance point process for stationary sequence. — Probab. Theory Rel. Fields, v. 78, 97–112.
• [57] Kabanov Yu.M., Liptser R.Sh. and Shiryaev A.N. (1983) Weak and strong convergence of the distributions of point processes. — Theory Probab. Appl., v. 28, No 2, 288–319.
• [58] Kabanov Yu.M. and Liptser R.Sh. (1983) On convergence in variation of the distributions of multivariate point processes. — Z. Wahrscheinlichkeitstheor. verw. Geb., v. 63, 475–485.
• [59] Kantorovich L.V. (1942) On the translocation of mass. — Doklady USSR Acad. Sci., v. 37, No 7–8, 227–229. Trans: Management Sci. (1958) v. 5, No 1, 1–4.
• [60] Kerstan J. (1964) Verallgemeinerung eines satzes von Prochorow und Le Cam. — Z. Wahrsch. Verw. Gebiete, v. 2, 173–179.
• [61] Khintchin A.Y. (1933) Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer.
• [62] Kolmogorov A.N. (1956) Two uniform limit theorems for sums of independent random variables. — Theory Probab. Appl., v. 1, No 4, 384–394.
• [63] Kontoyiannis I., Harremoes P. and Johnson O.T. (2005) Entropy and the law of small numbers. — IEEE Trans. Inform. Theory, v. 51, No 2, 466–472.
• [64] Kozulyaev P.A. (1939) Asymptotic analysis of a fundamental formula of Probability Theory. — Acad. Notes Moscow Univ., v. 15, 179–182.
• [65] Kruopis J. (1986) Precision of approximations of the generalized binomial distribution by convolutions of Poisson measures. — Lithuanian Math. J., v. 26, 37–49.
• [66] Leadbetter M.R., Lindgren G. and Rootzen H. (1983) Extremes and Related Properties of Random Sequences and Processes. New York: Springer Verlag.
• [67] LeCam L. (1960) An approximation theorem for the Poison binomial distribution. — Pacif. J. Math., v. 19, 1 3, p. 1181–1197.
• [68] LeCam L. (1965) On the distribution of sums of independent random variables. — In: Proc. Intern. Res. Sem. Statist. Lab. Univ. California, pp. 179–202. New York: Springer.
• [69] Liapunov A.M. (1901) Nouvelle forme du théor‘eme sur la limite des probabilites. — Mem. Acad. Imp. Sci. St.–Peterburg, v. 12, 1–24.
• [70] Logunov P.L. (1990) Estimates for the convergence rate to the Poisson distribution for random sums of independent indicators. — Theory Prob. Appl., 35, 587–590.
• [71] Loynes R.M. (1965) Extreme values in uniformly mixing stationary stochastic processes. — Ann. Math. Statist., v. 36, 993–999.
• [72] Major P. (1990) A note on the approximation of the uniform empirical process. — Ann. Probab., v. 18, No 1, 129–139.
• [73] Marcinkiewicz J. (1938) Sur les fonctions independantes II. — Func. Mat., v. 30, 349–364.
• [74] Matthes K., Kerstan J. and Mecke J. (1978) Infinitely divisible point processes. New York: Wiley.
• [75] Meshalkin L.D. (1960) On the approximation of polynomial distributions by infinitely-divisible laws. — Theory Probab. Appl., v. 5, No 1, 114–124.
• [76] Michel R. (1987) An improved error bound for the compound Poisson approximation of a nearly homogeneous portfolio. — ASTIN Bulletin, v. 17, 165–169.
• [77] Mihailov V.G. (2001) Estimate of the accuracy of compound Poisson approximation for the distribution of the number of matching patterns. — Theory Probab. Appl., v. 46, No 4, 667–675.
• [78] Mihailov V.G. (2008) Poisson-type limit theorem for the number of pairs of almost matching patterns. — Theory Probab. Appl., v. 53, No 1, 59–71.
• [79] Minakov A.A. (2015) Compound Poisson approximation of the distribution of the number of monotone runs of fixed length. — Appl. Discrete Math., v. 28, No 2, 21–29.
• [80] Mori T.F. (1990) More on the waiting time till each of some given patterns occurs as a run. — Can. J. Math., v. 42, No 5, 915–932.
• [81] Neuhauser C. (1994) A Poisson approximation for sequence comparisons with insertions and deletions. — Ann. Statist., v. 22, No 3, 1603–1629.
• [82] Novak S.Y. (2011) Extreme value methods with applications to finance. London: Chapman & Hall/CRC Press. ISBN 9781439835746
• [83] Novak S.Y. (2019) On the accuracy of Poisson approximation. — Extremes, v. 22.
• [84] Pittel B.G. (1981) Limiting behavior of a process of runs. — Ann. Probab., v. 9, no. 1, 119–129.
• [85] Presman E.L. (1983) Approximation of binomial distributions by infinitely divisible ones. — Theory Probab. Appl., v. 28, 393–403.
• [86] Presman E.L. (1985) The variation distance between the distribution of a sum of independent Bernoulli variables and the Poisson law. — Theory Probab. Appl., v. 30, No 2, 391–396.
• [87] Prokhorov Y.V. (1953) Asymptotic behavior of the binomial distribution. — Uspehi Matem. Nauk, v. 8, No 3(55), 135–142.
• [88] Prokhorov Y.V. (1956) Convergence of random processes and limit theorems in probability theory. — Theory Probab. Appl., v. 1, 157–214.
• [89] Rachev S.T. (1984) The Monge–Kantorovich problem on mass transfer and its stochastic applications. — Theory Probab. Appl., v. 29, No 4, 625–653.
• [90] Röllin A. (2005) Approximation of sums of conditionally independent variables by the translated Poisson distribution. — Bernoulli, v. 11, 1115–1128.
• [91] Romanowska M. (1977) A note on the upper bound for the distribution in total variation between the binomial and the Poisson distribution. — Statist. Neerlandica, v. 31, 127–130.
• [92] Roos M. (1994) Stein’s method for compound Poisson approximation: the local approach. — Ann. Appl. Probab., v. 4, No 4, 1177–1187.
• [93] Roos B. (1999) Asymptotic and sharp bounds in the Poisson approximation to the Poisson-binomial distribution. — Bernoulli, v. 5, No 6, 1021–1034.
• [94] Roos B. (2001) Sharp constants in the Poisson approximation. — Statist. Probab. Letters, v. 52, 155–168.
• [95] Roos B. (2003) Kerstan’s method for compound Poisson approximation. — Ann. Probab., v. 31, No 4, 1754–1771.
• [96] Roos B. (2003) Improvements in the Poisson approximation of mixed Poisson distributions. — J. Statist. Plan. Inference, v. 113, 467–483.
• [97] Roos B. (2015) Refined total variation bounds in the multivariate and compound Poisson approximation. — arXiv:1509.04167v1.
• [98] Ruzankin P.S. (2001) On the Poisson approximation of the binomial distribution. — Siberian Math. J., v. 42, No 2, 353–363.
• [99] Ruzankin P.S. (2004) On the rate of Poisson process approximation to a Bernoulli process. — J. Appl. Probab., v. 41, No 1, 271–276.
• [100] Ruzankin P.S. (2010) Approximation for expectations of unbounded functions of dependent integer-valued random variables. — J. Appl. Prob., v. 47, 594–600.
• [101] Salvemini T. (1943) Sul calcolo degli indici di concordanza tra due caratteri quantitativi. — Atti della VI Riunione della Soc. Ital. di Statistica.
• [102] Serfling R.J. (1975) A general Poisson approximation theorem. — Ann. Probab., v. 3, 726–731.
• [103] Serfling R.J. (1978) Some elementary results on Poisson approximation in a sequence of Bernoulli trials. — SIAM Rev., v. 20, No 3, 567–579.
• [104] Sevastyanov B.A. (1972) Limit Poisson law in a scheme of dependent random variables. — Theory Probab. Appl., v. 17, No 4, 733–737.
• [105] Schbath S. (2000) An overview on the distribution of word counts in Markov chains. — J. Comput. Biology, v. 7, 193–201.
• [106] Shevtsova I.G. (2011) On the absolute constants in the Berry–Esseen type inequalities for identically distributed summands. — arXiv:1111.6554v1.
• [107] Shorgin S.Y. (1977) Approximation of a generalized binomial distribution. — Theory Probab. Appl., v. 22, No 4, 846–850.
• [108] Smith R.L. (1988) Extreme value theory for dependent sequences via the Stein–Chen method of Poisson approximation. — Stoch. Proc. Appl., v. 30, No 2, 317–327.
• [109] Stein C. (1992) A way of using auxiliary randomization. — In: Probability Theory. Proc. Singapore Probab. Conf., pp. 159–180. Berlin: de Gruyter.
• [110] Tsaregradskii I.P. (1958) On uniform approximation of the binomial distribution with infinitely divisible laws. — Theory Probab. Appl., v. 3, No 4, 470–474.
• [111] Uspensky J.V. (1931) On Ch.Jordan’s series for probability. — Ann. Math., v. 32, No 2, 306–312.
• [112] Utev S.A. (1992) Extremal problems, characterisation, and limit theorems of Probability Theory. — DSc Thesis. Novosibirsk: Novosibirsk Inst. Math., 280 pp.
• [113] Vallander S.S. (1973) Calculation of the Wasserstein distance between probability distributions on the line. — Theory Probab. Appl., v. 18, No 4, 824–827.
• [114] Vasershtein L.N. (1969) Markov processes on a countable product of spaces describing large automated systems. — Probl. Inform. Trans., v. 14, 64–73.
• [115] Vervaat W. (1969) Upper bounds for the distance in total variation between the binomial or negative binomial and the Poisson distribution. — Statist. Neerlandica, v. 23, 79–86.
• [116] Witte H.-J. (1990) A unification of some approaches to Poisson approximation. — J. Appl. Probab., v. 27, No 3, pp. 611–621.
• [117] Xia A. (1997) On using the first difference in the Stein–Chen method. — Ann. Appl. Probab., v. 7, No 4, 899–916.
• [118] Xia A. (2005) Stein’s method and Poisson process approximation. — In: An introduction to Stein’s method (A.D. Barbour and L.H.Y. Chen, eds.) Singapore: World Scientific, 115–181.
• [119] Xia A. (2015) Stein’s method for conditional compound Poisson approximation. — Statist. Probab. Lett., v. 100, 19–26.
• [120] Yannaros N. (1991) Poisson approximation for random sums of Bernoulli random variables. — Statist. Probab. Lett., v. 11, 161–165.
• [121] Zacharovas V. and Hwang H.-K. (2010) A Charlier-Parseval approach to Poisson approximation and its applications. — Lith. Math. J., v. 50, No 1, 88–119.
• [122] Zaitsev A.Yu. (1983) On the accuracy of approximation of distributions of sums of independent random variables, which are non-zero with a small probability, by accompanying laws. — Theory Probab. Appl., v. 28, No 4, 657–669.
• [123] Zaitsev A.Yu. (1988) A multidimensional variant of Kolmogorov’s second uniform limit theorem. — Theory Prob. Appl., v. 34, 108–128.
• [124] Zaitsev A.Yu. (1991) An example of a distribution whose set of n-fold convolutions is uniformly separated from the set of infinitely divisible laws in the sense of the variation distance. — Theory Probab. Appl., v. 36, 419–425.
• [125] Zaitsev A.Yu. (2005) Approximation of a sample by a Poisson point process. — J. Math. Sciences, v. 128, No 1, 2556–2563.
• [126] Zubkov A.M. and Mihailov V.G. (1979) On the repetitions of $s$–tuples in a sequence of independent trials. — Theory Probab. Appl., v. 24, No 2, p. 267–279.
• [127] Zubkov A.M. and Serov A.A. (2012) A complete proof of universal inequalities for the distribution function of the binomial law. — Theory Probab. Appl., v. 57, No 3, 539–544.