Mod-φ convergence: Approximation of discrete measures and harmonic analysis on the torus
Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1115-1197.

In this paper, we relate the framework of mod-φ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is the one of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents of the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in permutations, number of prime divisors of a random integer, number of irreducible factors of a random polynomial, etc. Our approach allows us to deal with approximations in higher dimensions as well. In this setting, we bring out the influence of the correlations between the components of the random vectors in our asymptotic formulas.

Dans cet article, nous relions la théorie de la convergence mod-φ à la construction de schémas d’approximation pour des variables aléatoires à valeurs dans des réseaux. Le point de vue adopté est celui de l’analyse de Fourier dans l’algèbre de Wiener ; il permet le calcul d’équivalents asymptotiques des distances locales, de Kolmogorov et en variation totale. En utilisant des mesures signées au lieu de mesures de probabilités, nous construisons des approximations de distributions discrètes meilleures que l’approximation standard poissonnienne. Cette théorie s’applique à divers exemples issus de la combinatoire et de la théorie des nombres : nombre de cycles dans des permutations, nombre de diviseurs premiers d’un entier aléatoire, nombre de facteurs irréductibles d’un polynôme aléatoire, etc. Notre approche permet également des approximations en dimension supérieure. Dans ce cadre, nous mettons en évidence l’influence sur nos formules asymptotiques des corrélations entre les composantes des vecteurs aléatoires.

Received: 2018-03-19
Revised: 2019-03-18
Accepted: 2019-09-18
Published online: 2020-06-26
DOI: https://doi.org/10.5802/aif.3332
Classification: 62E17,  62E20,  60E10,  11N37,  11N45
Keywords: Mod-φ convergence, Wiener algebra, Lattice distributions, Approximation of random variables
@article{AIF_2020__70_3_1115_0,
     author = {Chhaibi, Reda and Delbaen, Freddy and M\'eliot, Pierre-Lo\"\i c and Nikeghbali, Ashkan},
     title = {Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {3},
     year = {2020},
     pages = {1115-1197},
     doi = {10.5802/aif.3332},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_3_1115_0/}
}
Chhaibi, Reda; Delbaen, Freddy; Méliot, Pierre-Loïc; Nikeghbali, Ashkan. Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus. Annales de l'Institut Fourier, Volume 70 (2020) no. 3, pp. 1115-1197. doi : 10.5802/aif.3332. https://aif.centre-mersenne.org/item/AIF_2020__70_3_1115_0/

[1] Arratia, Richard; Barbour, Andrew D.; Tavaré, Simon Logarithmic Combinatorial Structures: a Probabilistic Approach, EMS Monographs in Mathematics, European Mathematical Society, 2003 | Zbl 1040.60001

[2] Arratia, Richard; Goldstein, Larry; Gordon, Louis Two moments suffice for Poisson approximations: the Chen-–Stein method, Ann. Probab., Volume 17 (1989) no. 1, pp. 9-25 | Article | MR 972770 | Zbl 0675.60017

[3] Artin, Emil The gamma function, Athena Series. Selected Topics in Mathematics, Holt, Rinehart and Winston, 1964 | Zbl 0144.06802

[4] Barbour, Andrew D.; Čekanavičius, V. Total variation asymptotics for sums of independent integer random variables, Ann. Probab., Volume 30 (2002) no. 2, pp. 509-545 | MR 1905850 | Zbl 1018.60049

[5] Barbour, Andrew D.; Eagleson, G. K. Poisson approximation for some statistics based on exchangeable trials, Adv. Appl. Probab., Volume 15 (1983) no. 3, pp. 585-600 | Article | MR 706618 | Zbl 0511.60025

[6] Barbour, Andrew D.; Holst, Lars; Janson, Svante Poisson approximation, Oxford University Press, 1992 | Zbl 0746.60002

[7] Barbour, Andrew D.; Kowalski, Emmanuel; Nikeghbali, Ashkan Mod-discrete expansions, Probab. Theory Relat. Fields, Volume 158 (2014) no. 3-4, pp. 859-893 | Article | MR 3176367 | Zbl 1416.62131

[8] Barbour, Andrew D.; Xia, Aihua Poisson perturbations, ESAIM, Probab. Stat., Volume 3 (1999), pp. 131-150 | Article | Numdam | MR 1716120 | Zbl 0949.62015

[9] Bell, Eric Temple Partition polynomials, Ann. Math., Volume 29 (1927), pp. 38-46 | Article | MR 1502817 | Zbl 53.0132.02

[10] Betz, Volker; Ueltschi, Daniel Spatial random permutations and infinite cycles, Commun. Math. Phys., Volume 285 (2009) no. 2, pp. 469-501 | Article | MR 2461985 | Zbl 1155.82022

[11] Betz, Volker; Ueltschi, Daniel Spatial random permutations with small cycle weights, Probab. Theory Relat. Fields, Volume 149 (2011) no. 1-2, pp. 191-222 | Article | MR 2773029 | Zbl 1226.82003

[12] Betz, Volker; Ueltschi, Daniel; Velenik, Yvan Random permutations with cycle weights, Ann. Appl. Probab., Volume 21 (2011) no. 1, pp. 312-331 | Article | MR 2759204 | Zbl 1226.60130

[13] Borovkov, Konstantin A.; Pfeifer, Dietmar On improvements of the order of approximation in the Poisson limit theorem, J. Appl. Probab., Volume 33 (1996) no. 1, pp. 146-155 | Article | MR 1371962 | Zbl 0852.60025

[14] Le Cam, Lucien An approximation theorem for the Poisson binomial distribution, Pac. J. Math., Volume 10 (1960) no. 4, pp. 1181-1197 | Article | MR 142174 | Zbl 0118.33601

[15] Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo Representation theory and harmonic analysis of wreath products of finite groups, London Mathematical Society Lecture Note Series, Volume 410, London Mathematical Society, 2014 | MR 3202374 | Zbl 1318.20007

[16] Čekanavičius, V. Asymptotic expansions in the exponent: a compound Poisson approach, Adv. Appl. Probab., Volume 29 (1997) no. 2, pp. 374-387 | Article | MR 1450935 | Zbl 0895.60029

[17] Čekanavičius, V. Poisson approximations for sequences of random variables, Stat. Probab. Lett., Volume 39 (1998) no. 2, pp. 101-107 | Article | MR 1652516 | Zbl 0917.60025

[18] Čekanavičius, V.; Mikalauskas, M. Signed Poisson approximations for Markov chains, Stochastic Processes Appl., Volume 82 (1999) no. 2, pp. 205-227 | Article | MR 1700006 | Zbl 0997.60073

[19] Chen, Louis H. Y. On the convergence of Poisson binomial to Poisson distributions, Ann. Probab., Volume 2 (1974) no. 1, pp. 178-180 | Article | MR 370693 | Zbl 0276.60024

[20] Chen, Louis H. Y. Poisson approximation for dependent trials, Ann. Probab., Volume 3 (1975) no. 3, pp. 534-545 | Article | MR 428387 | Zbl 0335.60016

[21] Daley, Daryl J.; Vere-Jones, David An introduction to the theory of point processes. Vol. I: Elementary theory and methods, Probability and Its Applications, Springer, 2002

[22] Deheuvels, Paul; Pfeifer, Dietmar A semigroup approach to Poisson approximation, Ann. Probab., Volume 14 (1986) no. 2, pp. 663-676 | Article | MR 832029

[23] Delbaen, Freddy; Kowalski, Emmanuel; Nikeghbali, Ashkan Mod-φ convergence, Int. Math. Res. Not., Volume 11 (2015), pp. 3445-3485 | Zbl 1319.60040

[24] Ercolani, Nicolas M.; Ueltschi, Daniel Cycle structure of random permutations with cycle weights, Random Struct. Algorithms, Volume 44 (2012) no. 1, pp. 109-133 | Article | MR 3143592 | Zbl 1280.05004

[25] Erdős, Pál; Kac, Mark The Gaussian law of errors in the theory of additive number theoretic functions, Am. J. Math., Volume 62 (1940), pp. 738-742 | Article | MR 2374 | Zbl 0024.10203

[26] Féray, Valentin; Méliot, Pierre-Loïc; Nikeghbali, Ashkan Mod-φ convergence. Normality zones and precise deviations, SpringerBriefs in Probability and Mathematical Statistics, Springer, 2016 | Zbl 1387.60003

[27] Féray, Valentin; Méliot, Pierre-Loïc; Nikeghbali, Ashkan Mod-φ convergence, II: Estimates on the speed of convergence, Séminaire de Probabilités L (Lecture Notes in Mathematics) Volume 2252 (2019) | Article

[28] Flajolet, Philippe; Odlyzko, Andrew M. Random mapping statistics, Advances in Cryptology - EUROCRYPT 1989 (Lecture Notes in Computer Science) Volume 434 (1990), pp. 329-354 | MR 1083961 | Zbl 0747.05006

[29] Flajolet, Philippe; Odlyzko, Andrew M. Singularity analysis of generating functions, SIAM J. Discrete Math., Volume 3 (1990) no. 2, pp. 216-240 | Article | MR 1039294 | Zbl 0712.05004

[30] Flajolet, Philippe; Sedgewick, Robert Analytic Combinatorics, Cambridge University Press, 2009 | Zbl 1165.05001

[31] Harper, Adam J. Two new proofs of the Erdős–Kac theorem, with bound on the rate of convergence, by Stein’s method for distributional approximations, Proc. Camb. Philos. Soc., Volume 147 (2009) no. 1, pp. 95-114 | Article | MR 2507311 | Zbl 1195.11100

[32] Hwang, Hsien-Kuei Large deviations for combinatorial distributions. I. Central limit theorems, Ann. Appl. Probab., Volume 6 (1996) no. 1, pp. 297-319 | MR 1389841 | Zbl 0863.60013

[33] Hwang, Hsien-Kuei On convergence rates in the central limit theorems for combinatorial structures, Eur. J. Comb., Volume 19 (1998) no. 3, pp. 329-343 | Article | MR 1621021 | Zbl 0906.60024

[34] Hwang, Hsien-Kuei Asymptotics of Poisson approximation to random discrete distributions: an analytic approach, Adv. Appl. Probab., Volume 31 (1999) no. 2, pp. 448-491 | Article | MR 1724562 | Zbl 0945.60001

[35] Jacod, Jean; Kowalski, Emmanuel; Nikeghbali, Ashkan Mod-Gaussian convergence: new limit theorems in probability and number theory, Forum Math., Volume 23 (2011) no. 4, pp. 835-873 | MR 2820392 | Zbl 1225.15035

[36] Jiao, Ying; El Karoui, Nicole; Kurtz, David Gauss and Poisson approximations: application to CDOs pricing, J. Comput. Finance, Volume 12 (2008) no. 2, pp. 31-58 | Zbl 1175.91177

[37] Kahane, Jean-Pierre Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 50, Springer, 1970 | Zbl 0195.07602

[38] Katti, S. K. Infinite divisibility of integer-valued random variables, Ann. Math. Stat., Volume 38 (1967), pp. 1306-1308 | Article | MR 215333 | Zbl 0158.17004

[39] Katznelson, Yitzhak An introduction to harmonic analysis, Cambridge Mathematical Library, Cambridge University Press, 2004 | Zbl 1055.43001

[40] Kerstan, Johannes Verallgemeinerung eines Satzes von Prochorow und Le Cam, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 2 (1964), pp. 173-179 | Article | MR 165555 | Zbl 0123.35403

[41] Kowalski, Emmanuel; Nikeghbali, Ashkan Mod-Poisson convergence in probability and number theory, Int. Math. Res. Not., Volume 18 (2010), pp. 3549-3587 | Article | Zbl 1292.11104

[42] Kruopis, Julius Precision of approximations of the generalized binomial distribution by convolutions of Poisson measures, Lith. Math. J., Volume 26 (1986) no. 1, pp. 37-49 | Article | MR 847204 | Zbl 0631.60019

[43] Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, 1995 | Zbl 0824.05059

[44] Méliot, Pierre-Loïc Representation theory of symmetric groups, Discrete Mathematics and its Applications, CRC Press, 2017 | Article | Zbl 1418.20001

[45] Newman, Donald Joseph A simple proof of Wiener’s 1/f theorem, Proc. Am. Math. Soc., Volume 48 (1975) no. 1, p. 264-265 | MR 365002

[46] Nikeghbali, Ashkan; Zeindler, Dirk The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 4, pp. 961-981 | Article | Numdam | MR 3127909 | Zbl 1284.60171

[47] Petrov, Valentin Vladimirovich Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 82, Springer, 1975 | MR 388499

[48] Presman, Ernst L. Approximation of binomial distributions by infinitely divisible ones, Theory Probab. Appl., Volume 28 (1983) no. 2, pp. 393-403 | Article

[49] Prokhorov, Yurii V. Asymptotic behavior of the binomial distribution, Usp. Mat. Nauk, Volume 8 (1953) no. 3, pp. 135-142 | MR 56861 | Zbl 0051.10302

[50] Rényi, Alfréd; Turán, Pál On a theorem of Erdős–Kac, Acta Arith., Volume 4 (1958), pp. 71-84 | Article | Zbl 0083.26401

[51] Rhoades, Robert C. Statistics of prime divisors in function fields, Int. J. Number Theory, Volume 5 (2009) no. 1, pp. 141-152 | Article | MR 2499026 | Zbl 1179.11009

[52] Samuels, Stephen M. Positive-integer-valued infinitely divisible distributions (1975) (Purdue University Technical Report, Department of Statistics, Division of Mathematical Sciences, Mimeograph Series 406)

[53] Sato, Ken-Iti Lévy processes and infinitely divisible Ddstributions, Cambridge Studies in Advanced Mathematics, Volume 68, Cambridge University Press, 1999 | Zbl 0973.60001

[54] Steele, J. Michael Le Cam’s inequality and Poisson approximations, Am. Math. Mon., Volume 101 (1994) no. 1, pp. 48-54 | Article | MR 1252705 | Zbl 0802.60019

[55] Steutel, Fred W.; Van Harn, Klaas Infinite divisibility of probability distributions on the real line, Pure and Applied Mathematics, Volume 259, Marcel Dekker, 2004 | MR 2011862 | Zbl 1063.60001

[56] Szegö, Gábor Orthogonal polynomials, Colloquium Publications, Volume 23, American Mathematical Society, 1939 | Zbl 0023.21505

[57] Tenenbaum, Gérald Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics, Volume 46, Cambridge University Press, 1995 | MR 1342300 | Zbl 0831.11001