## The Annals of Applied Probability

### Large deviations for combinatorial distributions. I. Central limit theorems

Hsien-Kuei Hwang

#### Abstract

We prove a general central limit theorem for probabilities of large deviations for sequences of random variables satisfying certain analytic conditions. This theorem has wide applications to combinatorial structures and to the distribution of additive arithmetical functions. The method of proof is an extension of Kubilius' version of Cramér's classical method based on analytic moment generating functions. We thus generalize Cramér's and Kubilius's theorems on large deviations.

#### Article information

Source
Ann. Appl. Probab. Volume 6, Number 1 (1996), 297-319.

Dates
First available in Project Euclid: 18 October 2002

http://projecteuclid.org/euclid.aoap/1034968075

Digital Object Identifier
doi:10.1214/aoap/1034968075

Mathematical Reviews number (MathSciNet)
MR1389841

Zentralblatt MATH identifier
0863.60013

#### Citation

Hwang, Hsien-Kuei. Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab. 6 (1996), no. 1, 297--319. doi:10.1214/aoap/1034968075. http://projecteuclid.org/euclid.aoap/1034968075.

#### References

• 1 BENDER, E. A. 1973. Central and local limit theorems applied to asy mptotic enumeration. J. Combin. Theory Ser. A 15 91 111.
• 2 BERGERON, F., FLAJOLET, P. and SALVY, B. 1992. Varieties of increasing trees. In CAAP '92. Lecture Notes in Comput. Sci. 581 24 48. Springer, New York.
• 3 BUCKLEW, J. A. 1990. Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, New York.
• 4 CANFIELD, E. R. 1977. Central and local limit theorems for the coefficients of poly nomials of binomial ty pe. J. Combin. Theory Ser. A 23 275 290.
• 5 CANFIELD, E. R. 1980. Application of the Berry Esseen inequality to combinatorial estimates. J. Combin. Theory Ser. A 28 17 25.
• 6 CHERNOFF, H. 1952. A measure of asy mptotic efficiency for tests of a hy pothesis based on the sums of observations. Ann. Math. Statist., 23 493 507.
• 7 CRAMER, H. 1938. Sur un nouveau theoreme-limite de la theorie des probabilites. In ´ ´ ´ ´ Actualites Scientifiques et Industrielles 736 5 23. Herman, Paris. ´
• 8 DELANGE, H. 1971. Sur des formules de Atle Selberg. Acta Arith. 19 105 146.
• 9 ESSEEN, C.-G. 1945. Fourier analysis of distribution functions. A mathematical study of the Laplace Gaussian law. Acta Math. 77 1 125.
• 10 FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2, 2nd ed. Wiley, New York.
• 11 FLAJOLET, P., FRANC ¸ON, J. and VUILLEMIN, J. 1980. Sequence of operations analysis for dy namic data structures. J. Algorithms 1 111 141.
• 12 FLAJOLET, P. and ODLy ZKO, A. M. 1990. Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216 240.
• 13 FLAJOLET, P. and PRODINGER, H. 1986. Level number sequences for trees. Discrete Math. 65 149 156.
• 14 FLAJOLET, P. and SEDGEWICK, R. 1993. The average case analysis of algorithms, counting and generating functions. Research Report 1888, INRIA, Rocquencourt.
• 15 FLAJOLET, P. and SORIA, M. 1990. Gaussian limiting distributions for the number of components in combinatorial structures. J. Combin. Theory Ser. A 53 165 182.
• 16 FLAJOLET, P. and SORIA, M. 1993. General combinatorial schemes, Gaussian limit distributions and exponential tails. Discrete Math. 114 159 180.
• 17 GAO, Z. and RICHMOND, L. B. 1992. Central and local limit theorems applied to asy mptotic enumeration. IV. Multivariate generating functions. J. Comput. Appl. Math. 41 177 186.
• 18 HWANG, H.-K. 1994. Theoremes limites pour les structures combinatoires et les fonctions ´ arithmetiques. These, Ecole Poly technique. ´
• 19 HWANG, H.-K. 1995. On asy mptotic expansions in the central and local limit theorems for combinatorial structures. Unpublished manuscript.
• 20 HWANG, H.-K. 1995. On convergence rates in the central limit theorems for combinatorial structures. Unpublished manuscript.
• 21 HWANG, H.-K. 1995. Large deviations of combinatorial distributions. II. Local limit theorems. Unpublished manuscript. ¨
• 22 KHINTCHINE, A. Uber einen neuen Grennzwertsatz der Wahrscheinlichkeitsrechnung. Math. Ann. 101 745 752.
• 23 KNOPFMACHER, A., KNOPFMACHER, J. and WARLIMONT, R. 1992. Factorisatio numerorum'' in arithmetical semigroups. Acta Arith. 61 327 336.
• 24 KNOPFMACHER, J. 1979. Analy tic Arithmetic of Algebraic Function Fields. Lecture Notes in Pure Appl. Math. 50. Dekker, New York.
• 25 KOLASSA, J. E. 1994. Series Approximation Methods in Statistics. Lecture Notes in Statist. 88. Springer, New York.
• 26 KUBILIUS, J. 1964. Probabilistic Methods in the Theory of Numbers. Amer. Math. Soc., Providence, RI.
• 27 LUKACS, E. 1960. Characteristic Functions. Griffin, London.
• 28 MAEJIMA, M. and VAN ASSCHE, W. 1985. Probabilistic proofs of asy mptotic formulas for some orthogonal poly nomials. Math. Proc. Cambridge Philos. Soc. 97 499 510.
• 29 MAHMOUD, H. M. 1992. Evolution of Random Search Trees. Wiley, New York.
• 30 MEIR, A. and MOON, J. W. 1978. On the altitude of nodes in random trees. Canad. J. Math. 30 997 1015.
• 31 MEIR, A. and MOON, J. W. 1992. On nodes of given out-degree in random trees. In Fourth Czechoslovakian Sy mposium on Combinatorics, Graphs and Complexity. Annals of Z. Discrete Mathematics J. Nesetril and M. Fiedler, eds. 51 213 222. North-Holland, Amsterdam.
• 32 ODLy ZKO, A. M. 1992. Explicit Tauberian estimates for functions with positive coefficients. J. Comput. Appl. Math. 41 187 197.
• 33 PETROV, V. V. 1975. Sums of Independent Random Variables. Springer, Berlin. Trans. lated from the Russian by A. A. Brown.
• 34 RICHTER, W. 1964. A more precise form of an inequality of S. N. Bernstein. For large deviations. Selected Translations of Mathematical Statistics and Probability 4 225 232. Originally published in Vestnik Leningrad. Univ. Mat. Mek. Astronom. 14 Z. 24 29 1959.
• 35 SAULIS, L. and STATULEVICIUS, V. A. 1991. Limit Theorems for Large Deviations. Kluwer, Dordrecht.
• 36 SORIA, M. 1990. Methode d'analyse pour les constructions combinatoires et les algo´ ´ rithmes. These d'Etat, Univ. Paris-Sud.
• 37 STROOCK, D. W. 1984. An Introduction to the Theory of Large Deviations. Springer, New York.
• 38 SZEGO, G. 1977. Orthogonal Poly nomials. 4th ed. Amer. Math. Soc., Providence, RI. ¨
• 39 WHITTAKER, E. T. and WATSON, G. N. 1927. A Course of Modern Analy sis: An Introduction to the General Theory of Infinite Processes and of Analy tic Functions; with an Account of the Principal Transcendental Functions, 4th ed. Cambridge Univ. Press.