## Asymptotic Estimates of Elementary Probability Distributions (1996)

Venue: | Studies in Applied Mathematics |

Citations: | 12 - 6 self |

### BibTeX

@INPROCEEDINGS{Hwang96asymptoticestimates,

author = {Hsien-Kuei Hwang},

title = {Asymptotic Estimates of Elementary Probability Distributions},

booktitle = {Studies in Applied Mathematics},

year = {1996},

pages = {393--417}

}

### Years of Citing Articles

### OpenURL

### Abstract

Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.

### Citations

447 | Asymptotics and special functions - Olver - 1974 |

224 |
Discrete distributions
- Johnson, Kotz
- 1969
(Show Context)
Citation Context ...in the literature 1 due both to their intrinsic interest and, more importantly, to their wide applications to practical problems. We refer the reader to the recent monograph by Johnson, Kotz and Kemp =-=[25]-=-, and to Molenaar [31] for further information and references. In this paper, we shall introduce general analytic methods for deriving new estimates of elementary probability distribution functions. T... |

218 |
Sums of Independent Random Variables
- Petrov
- 1975
(Show Context)
Citation Context ...2]): \Pi m () = \Phi / m \Gammas+ 1 2 p ! + O i \Gamma1=2 j ; ass! 1, uniformly for m =s+ O i p j . For more precise Edgeworth expansions (with or without continuity correction and error bounds), see =-=[9, 12, 32, 34] and [25, -=-p. 162]. 2. Cram'er-type large deviations (cf. [32] [27, p. 100] [22, Ch. 3]): 1 \Gamma \Pi m () = (1 \Gamma \Phi(x)) exp ` \GammaH ` x p "` 1 +O ` x + 1 p " (m =s+ x p ); \Pi m () = \Phi(\G... |

180 |
Topics in poisson approximation
- Barbour
- 2001
(Show Context)
Citation Context ...onvolutions, cf. [25, Chs. 3 and 4]; and to other discrete distribution functions. Our techniques for deriving numerical bounds are also suitable for use for other Poisson approximation problems, cf. =-=[5, 38, 45]-=-. This paper is organized as follows. We first list some known asymptotic estimates concerning the Poisson distribution function in the next section. Then we state and prove our new results in Section... |

170 | Ramanujan's Notebooks , Part I - Berndt - 1985 |

155 | Probability Approximations via the Poisson Clumping Heuristic
- Aldous
- 1989
(Show Context)
Citation Context ...with more involved computations. 2.3 Poissonization Poissonization is a widely-used technique in stochastic process, summability of divergent sequence, analysis of algorithms, etc.; see, for example, =-=[1, 6, 18, 35, 19]-=-. The idea is roughly described as follows. Given a discrete probability distribution fa k g k0 (or, in general, a complex sequence), consider the Poisson generating function: b() = e \Gamma X j0 a j ... |

154 |
Asymptotic approximations of integrals
- Wong
- 1989
(Show Context)
Citation Context ...our of \Pi m () ass!1 and m runs through its possible values (depending on ). Whensis bounded and m!1, the asymptotic behaviour of \Pi m () can be easily derived by the usual saddle-point method, cf. =-=[10, 24, 26, 50]-=-. For completeness, we shall include the resulting formula at the end of x 2.2. 2.1 Known results Let us first list some known asymptotic estimates of \Pi m () in the literature. They are not intended... |

105 |
Saddlepoint Approximations in Statistics
- Daniels
- 1954
(Show Context)
Citation Context ...our of \Pi m () ass!1 and m runs through its possible values (depending on ). Whensis bounded and m!1, the asymptotic behaviour of \Pi m () can be easily derived by the usual saddle-point method, cf. =-=[10, 24, 26, 50]-=-. For completeness, we shall include the resulting formula at the end of x 2.2. 2.1 Known results Let us first list some known asymptotic estimates of \Pi m () in the literature. They are not intended... |

89 | Higher Transcendental Functions, Volume 1 - Magnus, Oberhettinger, et al. - 1955 |

85 | Saddlepoint Approximations
- Jensen
- 1995
(Show Context)
Citation Context ...our of \Pi m () ass!1 and m runs through its possible values (depending on ). Whensis bounded and m!1, the asymptotic behaviour of \Pi m () can be easily derived by the usual saddle-point method, cf. =-=[10, 24, 26, 50]-=-. For completeness, we shall include the resulting formula at the end of x 2.2. 2.1 Known results Let us first list some known asymptotic estimates of \Pi m () in the literature. They are not intended... |

85 | A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions - Whittaker, Watson - 1996 |

62 |
Saddlepoint approximations for the distribution of the sum of independent random variables
- Lugannani, Rice
- 1980
(Show Context)
Citation Context ...ould be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddle-point and a simple pole (one being allowed to approach the other), see =-=[47, 7, 36, 29, 11, 24]-=- and [50, pp. 356--360]. In particular, error bounds for (3) are discussed in [29, 39, 40]. 4. By the definition of \Pi m () \Pi m () = e \Gammasm m! X 0jm (m) j j ; (4) which is itself an asymptotic ... |

49 |
Théorèmes limites pour les structures combinatoires et les fonctions arithmetiques
- Hwang
- 1994
(Show Context)
Citation Context ...e integral. The underlying idea which consists of expanding the integrand at the saddle-point is a rather fruitful one and has been applied in many different contexts with satisfactory estimates (cf. =-=[40, 37, 44, 22]-=-). Besides the two classical distributions, our methods can also be applied to the many existing Poisson and binomial variants, mixtures, and convolutions, cf. [25, Chs. 3 and 4]; and to other discret... |

45 |
Introduction à la théorie analytique et probabiliste des nombres. Institut Elie Cartan
- Tenenbaum
- 1990
(Show Context)
Citation Context ...e integral. The underlying idea which consists of expanding the integrand at the saddle-point is a rather fruitful one and has been applied in many different contexts with satisfactory estimates (cf. =-=[40, 37, 44, 22]-=-). Besides the two classical distributions, our methods can also be applied to the many existing Poisson and binomial variants, mixtures, and convolutions, cf. [25, Chs. 3 and 4]; and to other discret... |

44 |
Introduction to Mathematical Probability
- Uspensky
- 1937
(Show Context)
Citation Context ...her complete account on different approximations and bounds for Bm (n) is given in [25, pp. 114-- 122], most of them being of normal-type; cf. also [2, 13]. To this account, we may add the references =-=[46, 5, 30, 43]-=-. 1. The classical de Moivre-Laplace theorem: Bm (n) = \Phi / m \Gamma np p npq ! +O i n \Gamma1=2 j ; the result being asymptotic for m = np + O ( p n). A precise error estimate was derived by Uspens... |

41 |
Tail probability approximations
- Daniels
(Show Context)
Citation Context ...ould be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddle-point and a simple pole (one being allowed to approach the other), see =-=[47, 7, 36, 29, 11, 24]-=- and [50, pp. 356--360]. In particular, error bounds for (3) are discussed in [29, 39, 40]. 4. By the definition of \Pi m () \Pi m () = e \Gammasm m! X 0jm (m) j j ; (4) which is itself an asymptotic ... |

32 |
Limiting distribution for the depth in PATRICIA tries
- Rais, Jacquet, et al.
- 1993
(Show Context)
Citation Context ...with more involved computations. 2.3 Poissonization Poissonization is a widely-used technique in stochastic process, summability of divergent sequence, analysis of algorithms, etc.; see, for example, =-=[1, 6, 18, 35, 19]-=-. The idea is roughly described as follows. Given a discrete probability distribution fa k g k0 (or, in general, a complex sequence), consider the Poisson generating function: b() = e \Gamma X j0 a j ... |

29 |
Handbook of the Poisson Distribution
- Haight
- 1967
(Show Context)
Citation Context ... in the literature. They are not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other types of approximations, see the monographs =-=[20, 31, 25, 5]-=-, the papers [2, 13] and the less known (in probability literature) paper by Norton [32] where a rather complete account (before 1976) on asymptotics of \Pi m () is given. Henceforth, \Phi(x) denotes ... |

29 |
Note on a paper by L
- Selberg
- 1954
(Show Context)
Citation Context ...e integral. The underlying idea which consists of expanding the integrand at the saddle-point is a rather fruitful one and has been applied in many different contexts with satisfactory estimates (cf. =-=[40, 37, 44, 22]-=-). Besides the two classical distributions, our methods can also be applied to the many existing Poisson and binomial variants, mixtures, and convolutions, cf. [25, Chs. 3 and 4]; and to other discret... |

24 |
Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function
- Temme
- 1975
(Show Context)
Citation Context ...(ms0); (2) where Q(a; ) = 1 \Gamma(a) Z 1 t a\Gamma1 e \Gammat dt; \Gamma being the gamma function. The asymptotic behaviour of Q has been extensively studied in the literature, most notably by Temme =-=[39, 40]-=-, see also [49]. For our purpose, let us mention the following expansion from [40] \Pi m\Gamma1 ()s1 \Gamma \Phi(j p 2m) \Gamma e \Gammamj 2 =2 p 2m X j0 b j m \Gammaj ; (3) as m !1 ands? 0, where r =... |

18 |
Marking in combinatorial constructions: Generating functions and limiting distributions
- Drmota, Soria
- 1995
(Show Context)
Citation Context ...lemma in Section 2.3), our expansions would hold in a wider range for the second parameter. This is so, for example, when f(z) = 1=\Gamma(z) in the case of the Stirling numbers of the first kind (cf. =-=[23, 14]-=-). A great deal of related combinatorial and arithmetical problems can be found in [22]. Integrals of the form 1 2i I z \Gammam\Gamma1 L(z) f(z)dz ( !1); with L(z) = X j1 z j j!(j \Gamma 1)! ; arising... |

17 | Asymptotic expansions for the stirling numbers of the first kind
- Hwang
- 1995
(Show Context)
Citation Context ... ;s4 (m) = m(m \Gamma 2) 8 ;s5 (m) = \Gamma m(5m \Gamma 6) 30 ;s6 (m) = \Gamma 3m 2 \Gamma 26m + 24 144 : The method of proof extends the original one by Selberg [37] to an asymptotic expansion as in =-=[23]-=-, the error term being further improved on here. Proof. By Cauchy's integral formula \Pi m () = e \Gamma 2i I jzj=i 1 1 \Gamma z z \Gammam\Gamma1 e z dz (0 ! i ! 1): (9) 5 Take i = r and expand the fa... |

16 |
Uniform asymptotic expansion of integrals with stationary point near algebraic singularity
- Bleistein
- 1966
(Show Context)
Citation Context ...ould be mentioned that (3) is also derivable by classical methods for uniform asymptotic expansions of integrals having a saddle-point and a simple pole (one being allowed to approach the other), see =-=[47, 7, 36, 29, 11, 24]-=- and [50, pp. 356--360]. In particular, error bounds for (3) are discussed in [29, 39, 40]. 4. By the definition of \Pi m () \Pi m () = e \Gammasm m! X 0jm (m) j j ; (4) which is itself an asymptotic ... |

13 |
Series Approximation Methods in Statistics
- Kolassa
- 1997
(Show Context)
Citation Context |

12 |
Uniform asymptotic expansion of Charlier polynomials
- Rui
- 1994
(Show Context)
Citation Context ...s0 (m) = 1;s1 (m) = 0: (13) These relations are computationally more useful than the defining equation (7). Thesj (m)'s are also related to Tricomi polynomials (cf. [42]) or Charlier polynomials (cf. =-=[5, 36, 8]-=-), see these cited papers and the references therein for asymptotics of this class of polynomials. 3. For the incomplete Gamma function, an expansion similar to (6) without explicit error bound was de... |

11 |
On the probability in the tail of a binomial distribution
- Littlewood
- 1969
(Show Context)
Citation Context ... the result is asymptotic for 0smsnp \Gamma OE(n), OE(n)= p n ! 1. His method is based on a continued fraction representation of Bm (n) and an approach by Markov (cf. [46, pp. 52--56]). 4. Littlewood =-=[28]-=- derived many asymptotic formulae for Bm (n) in different (overlapping) ranges of the interval 0smsnp. The results are too complicated to be listed here. His results were then corrected and extended b... |

10 |
The Analysis of Linear Probing Sort by the Use of a New Mathematical Transform
- Gonnet, Munro
- 1984
(Show Context)
Citation Context ...with more involved computations. 2.3 Poissonization Poissonization is a widely-used technique in stochastic process, summability of divergent sequence, analysis of algorithms, etc.; see, for example, =-=[1, 6, 18, 35, 19]-=-. The idea is roughly described as follows. Given a discrete probability distribution fa k g k0 (or, in general, a complex sequence), consider the Poisson generating function: b() = e \Gamma X j0 a j ... |

10 |
Simple Approximation to the Poisson, Binomial and Hypergeometric Distribution
- Molenaar
- 1971
(Show Context)
Citation Context ...e both to their intrinsic interest and, more importantly, to their wide applications to practical problems. We refer the reader to the recent monograph by Johnson, Kotz and Kemp [25], and to Molenaar =-=[31]-=- for further information and references. In this paper, we shall introduce general analytic methods for deriving new estimates of elementary probability distribution functions. These methods are best ... |

9 |
Uniform asymptotic expansions for saddle point integralsApplication to a probability distribution occurring in noise theory
- Rice
- 1968
(Show Context)
Citation Context |

8 |
Some Approximations to the Binomial Distribution Function
- Bahadur
- 1960
(Show Context)
Citation Context ...ve \Pi m () = e \Gammasm+1 m! 1s\Gamma m 1 + 1s\Gamma m \Gamma 1 1 + 1s\Gamma m \Gamma 2 1 + 1 . . . : Useful estimates for \Pi m () may be derived from this representation as in [46, pp. 53--56] and =-=[3]-=- for binomial distribution. 2. The polynomialssj (m) are related to Laguerre polynomials L (ff) n (x) = [z n ](1 \Gamma z) \Gammaff\Gamma1 exp (\Gammaxz=(1 \Gamma z)) bysj (m) = L (m\Gammaj) j (m); 7 ... |

8 |
Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions
- Temme
- 1990
(Show Context)
Citation Context |

5 |
Sur le nombre de facteurs premiers des entiers, Comptes Rendus de l’Académie des Sciences, Série I
- Balazard, Delange, et al.
- 1988
(Show Context)
Citation Context ...symptotic approximant" for more sophisticated problems. Besides the classical Poisson approximations (cf. [5]), let us mention the distribution of integerssx with a given number of prime factors =-=(cf. [4, 44]-=-) and the number of components in decomposable combinatorial structures (cf. [21]). Thus we shall investigate the asymptotic behaviour of \Pi m () ass!1 and m runs through its possible values (dependi... |

5 |
Searching for losers. Random Structures and Algorithms
- Grabner
- 1993
(Show Context)
Citation Context |

5 |
Uniform asymptotic expansions of Laplace integrals
- Temme
- 1983
(Show Context)
Citation Context ...108 (1 \Gamma r)( \Gamma m+ 1) 2 ( \Gamma m+ 2) 3 ( \Gamma m+ 3) 4 : Proof of Theorem 2. For convenience, let us write I(m; f) instead of \Pi m (), where f(z) = f 0 (z) = (1 \Gamma z) \Gamma1 . As in =-=[17, 41]-=-, our starting point is the formula I(m; f) = f(r 0 )e \Gammasm m! + e \Gamma 2i I jzj=r 0 z \Gammam\Gamma1 e z (f(z) \Gamma f(r 0 )) dz: By an integration by parts using (10), we have I(m; f) = f(r 0... |

4 |
The normal approximation to the Poisson distribution and proof of a conjecture of Ramanujan
- Cheng
- 1949
(Show Context)
Citation Context ...2]): \Pi m () = \Phi / m \Gammas+ 1 2 p ! + O i \Gamma1=2 j ; ass! 1, uniformly for m =s+ O i p j . For more precise Edgeworth expansions (with or without continuity correction and error bounds), see =-=[9, 12, 32, 34] and [25, -=-p. 162]. 2. Cram'er-type large deviations (cf. [32] [27, p. 100] [22, Ch. 3]): 1 \Gamma \Pi m () = (1 \Gamma \Phi(x)) exp ` \GammaH ` x p "` 1 +O ` x + 1 p " (m =s+ x p ); \Pi m () = \Phi(\G... |

4 |
On Littlewood’s estimate for the binomial distribution
- McKay
- 1989
(Show Context)
Citation Context ...her complete account on different approximations and bounds for Bm (n) is given in [25, pp. 114-- 122], most of them being of normal-type; cf. also [2, 13]. To this account, we may add the references =-=[46, 5, 30, 43]-=-. 1. The classical de Moivre-Laplace theorem: Bm (n) = \Phi / m \Gamma np p npq ! +O i n \Gamma1=2 j ; the result being asymptotic for m = np + O ( p n). A precise error estimate was derived by Uspens... |

4 |
Estimates for partial sums of the exponential series
- Norton
- 1978
(Show Context)
Citation Context ...of the second parameter m. For more information on other types of approximations, see the monographs [20, 31, 25, 5], the papers [2, 13] and the less known (in probability literature) paper by Norton =-=[32]-=- where a rather complete account (before 1976) on asymptotics of \Pi m () is given. Henceforth, \Phi(x) denotes the standard normal distribution function: \Phi(x) = 1 p 2 Z x \Gamma1 e \Gammat 2 =2 dt... |

3 |
A Normal Approximation for Beta and Gamma Tail Probabilities, Wahrscheinlichkeitstheorie 65
- Alfers, Dinges
- 1984
(Show Context)
Citation Context ... not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other types of approximations, see the monographs [20, 31, 25, 5], the papers =-=[2, 13]-=- and the less known (in probability literature) paper by Norton [32] where a rather complete account (before 1976) on asymptotics of \Pi m () is given. Henceforth, \Phi(x) denotes the standard normal ... |

3 |
Approximation of a generalized Binomial distribution, Theory Probab
- SHORGIN
- 1977
(Show Context)
Citation Context ...onvolutions, cf. [25, Chs. 3 and 4]; and to other discrete distribution functions. Our techniques for deriving numerical bounds are also suitable for use for other Poisson approximation problems, cf. =-=[5, 38, 45]-=-. This paper is organized as follows. We first list some known asymptotic estimates concerning the Poisson distribution function in the next section. Then we state and prove our new results in Section... |

3 |
Incomplete Laplace integrals: uniform asymptotic expansion with application to the incomplete beta function
- Temme
- 1987
(Show Context)
Citation Context ...her complete account on different approximations and bounds for Bm (n) is given in [25, pp. 114-- 122], most of them being of normal-type; cf. also [2, 13]. To this account, we may add the references =-=[46, 5, 30, 43]-=-. 1. The classical de Moivre-Laplace theorem: Bm (n) = \Phi / m \Gamma np p npq ! +O i n \Gamma1=2 j ; the result being asymptotic for m = np + O ( p n). A precise error estimate was derived by Uspens... |

2 |
Sur le nombre des diviseurs premiers de n
- Delange
- 1953
(Show Context)
Citation Context ...2]): \Pi m () = \Phi / m \Gammas+ 1 2 p ! + O i \Gamma1=2 j ; ass! 1, uniformly for m =s+ O i p j . For more precise Edgeworth expansions (with or without continuity correction and error bounds), see =-=[9, 12, 32, 34] and [25, -=-p. 162]. 2. Cram'er-type large deviations (cf. [32] [27, p. 100] [22, Ch. 3]): 1 \Gamma \Pi m () = (1 \Gamma \Phi(x)) exp ` \GammaH ` x p "` 1 +O ` x + 1 p " (m =s+ x p ); \Pi m () = \Phi(\G... |

2 |
Special cases of second order Wiener germ approximations, Probability Theory and Related Fields, 83:5--57
- Dinges
- 1989
(Show Context)
Citation Context ... not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other types of approximations, see the monographs [20, 31, 25, 5], the papers =-=[2, 13]-=- and the less known (in probability literature) paper by Norton [32] where a rather complete account (before 1976) on asymptotics of \Pi m () is given. Henceforth, \Phi(x) denotes the standard normal ... |

2 |
A convergent asymptotic representation for integrals
- Franklin, Friedman
- 1957
(Show Context)
Citation Context ...on can be repeated so that one would have an expansion whose successive terms are of order m j \Gamma2j (the terms in (6) are of order m [j=2] \Gammaj ). Adapting an idea due to Franklin and Friedman =-=[17]-=- for integrals of the form J = Z 1 0 ts\Gamma1 e \Gammat f(t) dt; we can answer the above question affirmatively. Theorem 2 Let r j = (m \Gamma j)=, js0. The distribution function \Pi m () satisfies t... |

2 |
A Poisson geometric law for the number of components in unlabelled combinatorial structures
- Hwang
- 1998
(Show Context)
Citation Context ...n approximations (cf. [5]), let us mention the distribution of integerssx with a given number of prime factors (cf. [4, 44]) and the number of components in decomposable combinatorial structures (cf. =-=[21]-=-). Thus we shall investigate the asymptotic behaviour of \Pi m () ass!1 and m runs through its possible values (depending on ). Whensis bounded and m!1, the asymptotic behaviour of \Pi m () can be eas... |

2 |
On uniform asymptotic expansions of definite integrals
- Wong
- 1973
(Show Context)
Citation Context ...a; ) = 1 \Gamma(a) Z 1 t a\Gamma1 e \Gammat dt; \Gamma being the gamma function. The asymptotic behaviour of Q has been extensively studied in the literature, most notably by Temme [39, 40], see also =-=[49]-=-. For our purpose, let us mention the following expansion from [40] \Pi m\Gamma1 ()s1 \Gamma \Phi(j p 2m) \Gamma e \Gammamj 2 =2 p 2m X j0 b j m \Gammaj ; (3) as m !1 ands? 0, where r = m=, j = sign(1... |

1 |
A class of polynomials related to those of Laguerre
- Temme
- 1985
(Show Context)
Citation Context ...j (m) \Gamma m j \Gamma1 (m) (js1);s0 (m) = 1;s1 (m) = 0: (13) These relations are computationally more useful than the defining equation (7). Thesj (m)'s are also related to Tricomi polynomials (cf. =-=[42]-=-) or Charlier polynomials (cf. [5, 36, 8]), see these cited papers and the references therein for asymptotics of this class of polynomials. 3. For the incomplete Gamma function, an expansion similar t... |

1 |
On Ch. Jordan's series for probability
- Uspensky
- 1931
(Show Context)
Citation Context ...onvolutions, cf. [25, Chs. 3 and 4]; and to other discrete distribution functions. Our techniques for deriving numerical bounds are also suitable for use for other Poisson approximation problems, cf. =-=[5, 38, 45]-=-. This paper is organized as follows. We first list some known asymptotic estimates concerning the Poisson distribution function in the next section. Then we state and prove our new results in Section... |

1 |
der Waerden, On the method of saddle-points
- van
- 1951
(Show Context)
Citation Context |