LARGE DEVIATIONS OF COMBINATORIAL ... - Project Euclid

3 downloads 0 Views 155KB Size Report
receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results ..... where Cz is logarithmic with parameters ζ a K b and Qw z satisfies.
The Annals of Applied Probability 1998, Vol. 8, No. 1, 163–181

LARGE DEVIATIONS OF COMBINATORIAL DISTRIBUTIONS II. LOCAL LIMIT THEOREMS By Hsien-Kuei Hwang1 Academia Sinica We derive a general local limit theorem for probabilities of large deviations for a sequence of random variables by means of the saddlepoint method on Laplace-type integrals. This result is applicable to parameters in a number of combinatorial structures and the distribution of additive arithmetical functions.

1. Main result. This paper is a sequel to our paper [18] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions. The ranges of large deviations treated here are usually referred to as “moderate deviations” in probability literature (cf., e.g., [23]); we follow the same terms as in Part I [18] to keep consistency. In this paper we consider corresponding local limit theorems. More precisely, given a sequence of integral random variables ”n •n≥1 each of maximal span 1 (see below for definition), we are interested in the asymptotic behavior of the probabilities Pr”n = m•;

m ∈ N; m = µn ± xn σn ; µn x= E n ; σn2 x= Var n ‘;

as n → ∞, where xn can tend to ∞ with n at a rate that is restricted to Oσn ‘. Our interest here is not to derive asymptotic expression for Pr”n = m• valid for the widest possible range of m, but to show that for m lying in the interval µn ± Oσn2 ‘, very precise asymptotic formulas can be obtained. These formulas are in close connection with our results in [18]. Although local limit theorems receive a constant research interest [2, 3, 7, 14, 13, 24], our approach and results, especially Theorem 1, seem rarely discussed in a systematic manner. Recall that a lattice random variable X is said to be of maximal span h if X takes only values of the form b + hk, k ∈ Z, for some constants b and h > 0, and there does not exist b0 and h0 > h such that X takes only values of the form b0 + h0 k. Let us now state our main result. Let ”n •n≥1 be a sequence of random variables taking only integral values. Suppose that n is of maximal span 1 for n ≥ n0 n0 ≥ 1‘. P Assume further that the moment generating functions Mn s‘ x= E en s = m∈Z Pr”n = m•ems satisfy (1)

Mn s‘ = eφn‘us‘+vs‘ 1 + Oκ−1 n ‘‘;

n → ∞;

Received January 1995; revised June 1997. partially supported by ESPRIT Basic Research Action No. 7141 (ALCOM II) while ´ the author was at Ecole Polytechnique. AMS 1991 subject classifications. Primary 60F10; secondary 05A16, 11K65. Key words and phrases. Large deviations, local limit theorems, asymptotic expansion, saddlepoint method, combinatorial schemes, singularity analysis, additive arithmetical functions. 1 Research

163

164

H.-K. HWANG

uniformly for ŽsŽ ≤ ρ, s ∈ C, ρ > 0, where the following hold: 1. ”φn‘• is a sequence of n such that φn‘ → ∞ as n → ∞; 2. us‘ and vs‘ are functions of s independent of n and are analytic for ŽsŽ ≤ ρ; furthermore u00 0‘ 6= 0; 3. κn → ∞; 4. the moment generating functions Mn s‘ satisfy condition (A): there exist constants 0 < ε ≤ ρ and c = cε; r‘ > 0, where ε > 0 may be taken arbitrarily small but fixed, such that Mn r + it‘  −cφn‘ A‘ ; M r‘ = O e n uniformly for −ρ ≤ r ≤ ρ and ε ≤ ŽtŽ ≤ π, as n → ∞.

As in [18], let us introduce the following notation: um x= um‘ 0‘; µn x= u1 φn‘;

vm x= vm‘ 0‘; σn2 x= u2 φn‘:

m = 1; 2; 3; : : : ;

The symbol ’zn “fz‘ denotes the coefficient of zn in fz‘.

p Theorem 1. If m = µn +xσn , x = o φn‘‘, then the probabilities Pr”n = m• satisfy asymptotically  2 X 5k x‘ e−x /2+φn‘Qξ‘ 1+ Pr”n = m• = p u2 φn‘‘k/2 2πu2 φn‘ 1≤k≤ν (2)   ŽxŽ + 1 ŽxŽν+1 + 1 ; + p +O φν+1‘/2 n‘ κn φn‘ P where ξ = x/σn , Qξ‘ = Quy ξ‘ = k≥3 qk ξk is analytic at 0 with the coefficients qk defined by   0 −1 k−2 00 u w‘ − u1 −k qk = (3) ; k = 3; 4; 5; : : : ; ’w “u w‘ k u2 w ν is a nonnegative integer (depending upon the error term κn−1 ) and the 5k x‘ are polynomials of degree k such that 52j x‘ has only even powers of x and 52j−1 x‘ has only odd powers of x, for j = 1; 2; : : : : Note that the Cram´er-type power series Qy‘ is exactly the same as in Theorem 1 of [18]. This theorem generalizes a result of Richter [28] on local limit theorem for large deviations for sums of independent and identically distributed random variables. Richter’s result has been generalized by many authors; see [4, 21] and the references therein. In a combinatorial context, general (multidimensional) local limit theorems have been derived by Gao and Richmond [13] under different settings. Their

LOCAL LIMIT THEOREMS

165

methods, including the Fourier inversion formula and the technique of “shifting the mean,” although apparently different, have the same analytic character. On the other hand, our results are comparatively more precise. The asymptotic relation (1) is a priori a local one and need not be uniformly valid throughout the region −ρ ≤ 0, such that < ur + it‘ − ur‘ < −δ;

−ρ ≤ r ≤ ρ; ε ≤ ŽtŽ < π:

Such a condition is used in [13]. Note that the remaining cases t = ±π will not cause further difficulty due to the uniform continuity of characteristic functions. In general, the availability of condition (A), usually a result of aperiodicity of coefficients, separates, on the one hand, the central limit theorem from the local limit theorem for lattice random variables. It also has, on the other hand, a straightforward effect on estimates of probabilities of large deviations, as may be simply seen by the following rough arguments which can be justified using similar (and simpler) proof techniques of Theorem 1. By the integral formula (5) below, we have the upper bound e−mr Z π Pr”n = m• ≤ ŽMn r + it‘Ž dt: 2π −π

If condition (A) is not available, we have

 Pr”n = m• ≤ e−mr Mn r‘ = O e−mr+ur‘φn‘ y

on the other hand, assuming condition (A) [cf. (7) below],   Zε −mr+ur‘φn‘ φn‘ 0 [in a way to minimize the value of e−mr+ur‘φn‘ ]. If m = aφn‘ and the real solution r0 to the equation u0 s‘ = a satisfies −ρ ≤ r0 ≤ ρ, then we take r = r0 ; otherwise, we take r = ρ so that the right-hand side of (4) is uniformly small. Therefore, the smooth (or regularity) condition (A), which reflects the concentration of moment generating functions around real axis, yields the correct order of tail probabilities. Estimates for Pr”n ≥ m• for Žm/φn‘ − u1 Ž > ε > 0 are easily derived by summation of the right-hand side of (4). We prove Theorem 1 in the next section; the method of proof is based on the saddlepoint method on Laplace-type integrals. Many immediate consequences of this result will be given in Section 3. These results will then be applied to the combinatorial schemes of Flajolet and Soria [11, 12] in Section 4. Finally, we discuss some examples in Section 5. We conclude this paper with an extension of our results.

166

H.-K. HWANG

Throughout this paper, all generating functions (ordinary, exponential, bivariate, etc.) denote functions analytic at 0 with nonnegative coefficients. Following a number-theoretic convention, the symbols O and  are equivalent and will be used interchangeably as is convenient. All limits, (including O, , o and ∼), whenever unspecified, will be taken as n → ∞. The symbols ε; δ always represent sufficiently small (but fixed, namely, independent of the major asymptotic parameter), positive numbers whose values may differ from one occurrence to another. p 2. Proof of Theorem 1. Write m = µn + xσn , x = O φn‘‘. We start from the integral formula 1 Z r+iπ Mn z‘e−mz dz =x In; m ; (5) −ρ ≤ r ≤ ρ: Pr”n = m• = 2πi r−iπ Divide the integral In; m into two parts: In; m = I1 + I2 , where 1 Z 1 Zε M r + it‘e−mr−mit dt: Mn r + it‘e−mr−mit dty I2 = I1 = 2π −ε 2π ε 0, we can carry out the change of variable y = λn t = p u00 r‘φn‘ t and obtain      Z ελn X 1 iy k k‘ uk+2‘ r‘iy‘2 −y2 /2 −1 dy: v r‘ + e exp I3 = λn k! λn k + 1‘k + 2‘u00 r‘ −ελn k≥1

Define polynomials Pk z‘ by the formal expansion (cf. [22], page 149; [16], Section 2.4)      X 1 z k k‘ X Pk z‘ uk+2‘ r‘z2 exp = v r‘ + ; 00 r‘ k! λ k + 1‘k + 2‘u λkn n k≥1 k≥0

the degree of Pk z‘ being equal to 3k k ≥ 0‘. The same argument as that used in [16], Lemma 2 (cf. also [6], Lemma 2, Chapter VII; [22], page 151) leads to     X 1 Z∞ y2 1 I3 = exp − Pk iy‘ dy + O ν+2 ; k+1 −∞ 2 λn 0≤k≤ν λn

for any nonnegative integer ν. Now from the explicit expression of Pk z‘ (cf. [16], page 67)  jl Y X ul+2‘ r‘z2 1 l‘ k v r‘ + ; Pk z‘ = z l!jl ! l + 1‘l + 2‘u00 r‘ j +2j +···+kj =k 1≤l≤k 1

2

k

j1 ;j2 ;:::;jk ≥0

k = 1; 2; 3; : : : ;

we observe that the P2k z‘ contain only even powers of z (from z2k ; z2k+2 ; : : : ; z6k ) and that the P2k+1 z‘ contain only odd powers of z (from z2k+1 ; z2k+3 ; : : : ; R∞ 2 z6k+3 ) for k ≥ 0. It follows that −∞ e−y /2 P2k+1 iy‘ dy = 0 for all nonnegative integer k. Thus we obtain the asymptotic expansion √   X 2π d2k r‘ (10) ; I3 ∼ p 1+ u00 r‘φn‘‘k u00 r‘φn‘ k≥1

168

H.-K. HWANG

where (ckj r‘ x= ’z2j “P2k z‘),

1 Z ∞ −y2 /2 d2k r‘ = √ e P2k iy‘ dy 2π −∞ X 2j‘! −1‘j j ckj r‘; = 2 j! k≤j≤3k

k = 1; 2; 3; : : : :

Returning to Pr”n = m•, we have, by (5)–(10),  X eur‘φn‘+vr‘−mr d2k r‘ Pr”n = m• = p 1+ 00 r‘φn‘‘k 00 u 2πu r‘φn‘ 1≤k≤ν (11)   ŽrŽ 1 1 + +O + ; p φν+1 n‘ κn φn‘ κn φn‘ for some nonnegative integer ν. In particular, d2 r‘ =

v0 r‘u000 r‘ 5u000 r‘2 1 v00 r‘ v0 r‘2 + − − − : 8 2 2 2u4‘ r‘ 24u4‘ r‘2

Note that when x = 0 (this implies that the µn are integers), we obtain (r = 0)    X 1 d2k 0‘ 1 1 Pr”n = µn • = p 1+ + ; + O φν+1 n‘ κn φn‘ u2 φn‘‘k 2πu2 φn‘ 1≤k≤ν

(probability of the mode). Let us now prove (2). Set ξ = x/σn . In [18], we showed that φn‘ur‘ − ru0 r‘‘ = −

x2 + φn‘Qξ‘; 2

and similarly, we can write 1 evr‘ d2k r‘ = k+1/2 00 k+1/2 u r‘ u2



X

j≥0

 bkj ξj ;

k = 0; 1; 2; : : : ;

for some coefficients bkj . Theorem 1 follows from collecting terms according to the powers of ξ (or x): X X bk−j; 2j+1 x2j+1 ; bk−j; 2j x2j ; and 52k+1 x‘ = 52k x‘ = 0≤j≤k

0≤j≤k

for k = 0; 1; 2; : : : : In particular,   u 50 x‘ = 1; 51 x‘ = v1 − 3 x; 2u2   2 u4 1 v2 v1 v1 u3 v21 v1 u3 5u23 5u23 v2 2 − : x + + − − − + + − 52 x‘ = 2 2 u2 4u2 8 2 2 2u4 8u22 24u24 This completes the proof. 2

LOCAL LIMIT THEOREMS

169

3. Some corollaries of Theorem 1. The following two corollaries are immediate consequences of Theorem 1. If m = µn + xσn , x = oφn‘1/2 ‘, then    2 e−x /2+φn‘Qξ‘ ŽxŽ + 1 Pr”n = m• = p 1+O p : 2πu2 φn‘ φn‘

Corollary 1.

If m = µn + xσn , x = oφn‘1/6 ‘, then    3 2 ŽxŽ + 1 e−x /2 : 1+O p Pr”n = m• = p 2πu2 φn‘ φn‘

Corollary 2.

Other corollaries like those in Section 3 of [18] can be derived. As in [18], define the real function 3 of y, y > −1, by 3y‘ x= 1 + y‘ log1 + y‘ − y − the series being convergent for ŽyŽ ≤ 1.

X −1‘k y2 = yk ; 2 kk − 1‘ k≥3

Theorem 2. Let us‘ = es − 1 in Theorem 1 pand assume that κn−1 = p oφn‘−M ‘ for any M > 0. Then, for m = φn‘ + x φn‘, x = o φn‘‘, we have p exp−x2 /2 − φn‘3x/ φn‘‘‘ Pr”n = m• = p 2πφn‘ (12)    X 5k x‘ ŽxŽν+1 + 1 × 1+ ; +O φn‘k/2 φn‘ν+1‘/2 1≤k≤ν

or, equivalently, the Poisson approximation formula    X $k x‘ φn‘m −φn‘ ŽxŽν+1 + 1 1+ e (13) ; Pr”n = m• = + O m! φn‘k/2 φn‘ν+1‘/2 1≤k≤ν for some polynomials $k x‘ and some nonnegative integer ν.

Proof. For (12), the saddle point r in (8) satisfies er = 1 + ξ = mφn‘‘−1 . For (13), we have, by (11),    X d2k r‘ e−φn‘+m+vr‘ φn‘m 1 Pr”n = m• = 1+ + O √ mν+1 mk 2πm mm 1≤k≤ν    X de2k r‘ e−φn‘ φn‘m vr‘ 1 1+ e ; = +O m! mν+1 mk 1≤k≤ν

by applying Stirling’s formula backwards. The required result follows from expanding each factor evr‘ de2k r‘/mk in powers of ξ. 2

170

H.-K. HWANG

In particular, $2 x‘ = 12 v21 + v2 − v1 ‘x2 − 1‘

$1 x‘ = v1 x;

and in general the $k x‘ have the same degree property as 5k x‘. From the Poisson approximation formula (13), we can easily derive precise asymptotics for, say, the total variation distance of n and a Poisson distribution with mean φn‘. Also our Theorem 1 is useful for further asymptotics of different probability metrics; compare, for example, [1] and the references cited there. 4. Combinatorial schemes of Flajolet and Soria. In this section, we apply the theorems derived in previous sections to the combinatorial distributions studied by Flajolet and Soria [11, 12]. These distributions are classified according to the type of singularity of their bivariate generating functions. 4.1. The exp–log class. We state a definition for logarithmic function slightly stronger than that in [12] and [18]. A generating function Cz‘ is called logarithmic (cf. [11]) if (i) Cz‘ is analytic for z ∈ 1: 1 x= ”zx ŽzŽ ≤ ζ + ε

and

Ž argz − ζ‘Ž ≥ δ•\”ζ•;

ε > 0; 0 < δ < π/2;

ζ > 0 being the radius of convergence and the sole singularity of C for z ∈ 1, and (ii) there exists a constant a > 0, such that for z ∼ ζ, z ∈ 1, Cz‘ = a log

1 + H1 − z/ζ‘1/b ‘; 1 − z/ζ

where b = 1; 2; 3; : : :, Hu‘ is analytic at 0 and satisfies the expansion X Hu‘ = K + hk uk ; k≥1

convergent for ŽuŽ ≤ ε, K being some constant. For brevity, we say that C is logarithmic with parameters ζ; a; K; b‘. Now consider generating functions of the form X Pn w‘zn = ewCz‘ Qw; z‘; Pw; z‘ = n; m≥0

where Cz‘ is logarithmic with parameters ζ; a; K; b‘ and Qw; z‘ satisfies the following two conditions:

1. As a function of z, Qw; z‘ is analytic for ŽzŽ ≤ ζ, namely, it has a larger radius of convergence than C. 2. As a function of w, Qw; ζ‘ is analytic for ŽwŽ ≤ η, where η > 1. Roughly, these assumptions imply that for any fixed w, ŽwŽ ≤ η, Pw; z‘ satisfies    z −aw 1 + O 1 − z/ζ‘1/b ; z ∼ ζ; z 6∈ ’ζ; ∞‘; Pw; z‘ = eKw Qw; ζ‘ 1 − ζ

171

LOCAL LIMIT THEOREMS

and Pw; z‘ is analytically continuable to a 1-region. We can then apply the singularity analysis of Flajolet and Odlyzko [10] to deduce the asymptotic formula (14)

Pn w‘ x= ’zn “Pw; z‘ =

 ζ −n naw−1 Kw e Qw; ζ‘ 1 + O n−1/b ; 0aw‘

the O-term being uniform with respect to w, ŽwŽ ≤ η. We note that although an asymptotic expansion in terms of the ascending powers of n−1/b can be derived under our assumptions, the expression (14) suffices for our purposes. Since η > 1, Pn 1‘ is well defined. Thus for the moment generating functions Mn s‘ of the random variables n defined by Mn s‘ x= E en s = Pn es ‘/Pn 1‘, we have Mn s‘ =

Kes −1‘ 0a‘Qes ; ζ‘ es −1‘a log n e e 0aes ‘Q1; ζ‘

 1 + O n−1/b ;

uniformly for − log η ≤ 0. The application of Theorem 2 is straightforward since from (14) the Pn w‘ are aperiodic for sufficiently large n and condition (A) is easily checked. p Theorem 3. Let  be defined as above. If m = a log n + x a log n, x = n p o log n‘, then we have p exp−x2 /2 − a log n · 3x/ a log n‘‘ Pr”n = m• = p 2πa log n    X 5k x‘ ŽxŽν+1 + 1 × 1+ : + O a log n‘k/2 log n‘ν+1‘/2 1≤k≤ν

In other words, n obeys asymptotically a Poisson distribution of parameter a log n: Pr”n = m• =

   X $k x‘ ŽxŽν+1 + 1 a log n‘m −a log n 1+ e : + O m! a log n‘k/2 log n‘ν+1‘/2 1≤k≤ν

Proof. Take φn‘ = a log n in Theorem 2. 2 4.2. The algebraic–logarithmic class. Next, let us consider generating functions of the form X Pn w‘zn Pw; z‘ = n; m≥0

(15)

=

β  1 1 log ; 1 − wCz‘‘α 1 − wCz‘

C0‘ = 0;

172

H.-K. HWANG

where β ∈ N, α ≥ 0, and α + β > 0. Define the moment generating functions Mn s‘ of the random variables n by the coefficients of Pn w‘: (16)

Mn s‘ =

X

m≥0

Pr”n = m• ems =

’zn “Pes ; z‘ : ’zn “P1; z‘

Definition (1-regular function [12]). A generating function Cz‘ 6≡ zq (q = 0; 1; 2; : : :) analytic at z = 0 is called 1-regular if there exists a positive number ρ < ζ, ζ being the radius of convergence of Cz‘, such that Cρ‘ = 1. Assume, without loss of generality, that Cz‘ is aperiodic, namely, P Cz‘ 6≡ ze n≥0 cn znd for some integers e ≥ 0 and d ≥ 2. (This aperiodic condition is slightly stronger than the one in [12], page 166.) We need the following simple lemma. Lemma 1. Let Pn w‘ = ’zn “Pw; z‘, where Pw; z‘ has the form (15). Cz‘ is aperiodic iff Pn w‘ is aperiodic for n ≥ n0 , n0 depending only upon β and the period of C. Proof. If Cz‘ = zq Dzp ‘, where q ≥ 0, p ≥ 1 are integers and D is a power series, then X Pn w‘ = Ak Bkj wk ; n ≥ 1; qk+pj=n k; j≥0

where  Ak = ’z “ k

1 1−z

α 

1 log 1−z



and

Bkj = ’zpj “Dzp ‘k :

Since Ak > 0 for k ≥ β, the result follows from considering the cases p = 1 (C being aperiodic) and p ≥ 2 (C being periodic). 2 Assume that Cz‘ is 1-regular and ρw‘ satisfies ρ1‘ = ρ and Cρw‘‘ = w−1 for Žw − 1Ž ≤ ε. By our periodicity condition, implicit function theorem and singularity analysis, we obtain (cf. [10, 12, 19]), (17)

Pn w‘ =

 ρw‘−n nα−1 log n‘β 1 + εα; ⠐n‘ ; 0 α wρw‘C ρw‘‘‘

uniformly for Žw − 1Ž ≤ ε, where ( n−1 ; if α > 0 and β = 0‘ or α = 0 and β = 1‘; εα; ⠐n‘  −1 log n‘ ; if α = 0 and β ≥ 2‘ or α > 0 and β > 0‘: This result gives the local behavior of Pn w‘ for w ∼ 1. To apply Theorem 1, we need an estimate for ŽPn reit ‘Ž for r near 1 and ε ≤ ŽtŽ ≤ π.

173

LOCAL LIMIT THEOREMS

A uniform estimate for Pn w‘. Observe first that ρw‘ satisfies Žρreit ‘Ž ≥ ρr‘ for r and t in the region of validity of (17), in particular, for Žreit − 1Ž ≤ ε. For, otherwise, using the fact that the coefficients of Cz‘ are nonnegative, r−1 = ŽCρreit ‘‘Ž ≤ CŽρreit ‘Ž‘ < Cρr‘‘ = r−1 ; a contradiction. We deduce, by 1-regularity of Cz‘, that Pn reit ‘ 2 (18) P r‘  exp−qr‘nt ‘; n

holds uniformly for Žr − 1Ž ≤ δ and ŽtŽ ≤ ε, where qr‘ > 0 is sufficiently small. We now show that such an estimate subsists for ε ≤ ŽtŽ ≤ π, namely, Mn s‘ satisfies condition (A). Lemma 2. If Cz‘ is 1-regular and α ≥ 0, β ∈ N satisfies α + β > 0, then there exists an absolute constant q = qr‘ > 0, independent of n; t; α; β, such that, for n sufficiently large, Pn r expit‘‘  Pn r‘ exp−qr‘nt2 ‘;

(19)

holds uniformly for 1 − δ ≤ r ≤ 1 + δ and −π ≤ t ≤ π. Proof. The case ŽtŽ ≤ ε having been proved in (18), we consider the remaining range ε ≤ ŽtŽ ≤ π. Consider first the case α > 0 and β = 0. We use induction on n. By Lemma 1, there exists a constant n0 > 1 independent of α such that Pn w‘ is aperiodic for n ≥ n0 . It is easily seen, by distinguishing two cases: ŽtŽ ≤ ε and ε ≤ ŽtŽ ≤ π, that (19) is satisfied [qr‘ can be chosen small enough so that it depends only upon C but not upon α]. Now suppose that (19) holds for n0 ≤ n ≤ N−1. Differentiating the defining equation (15) of Pn w‘ and multiplying both sides by the factor z1 − wCz‘‘ yields the recurrence relation X NPN w‘ = w ck PN−k w‘αk + N − k‘; (20) N ≥ 1; 1≤k≤N

with P0 w‘ = 1, where ck x= ’zk “Cz‘. Set w = reit , where 1 − δ ≤ r ≤ 1 + δ and ε ≤ ŽtŽ ≤ π. We divide the estimate of the sum in (20) into two parts separated at k = †N/2‡. For †N/2‡ < k ≤ N, we use (17) in the form PN r‘  ρr‘−N Nα−1 ;

(21) and obtain X reit

†N/2‡ 1, where     Y 1 z 1 z Vz‘ = 1+ ; 1− 0z‘ p x prime p p−1 the product being extended over p p. Applying Theorem 2, p all prime numbers we obtain, for m = log log n + x log log n, x = o log log n‘, p exp−x2 /2 − log log n · 3x/ log log n‘‘ Pr”ξn = m• = p 2π log log n    X 5k x‘ ŽxŽν+1 + 1 × 1+ ; +O log log n‘k/2 log log n‘ν+1‘/2 1≤k≤ν

a formula that can be deduced by a result of Selberg [29] (cf. also [8], [25] and [30], Theorem II.6.4) but does not seem to have been stated in this form. The same results hold for n‘, the total number of prime divisors of n (with multiplicities), and for many other arithmetic functions; see [18].

6. Extension. From the proof of Theorem 1 [see formula (11)], it is obvious that we have in fact proved more, namely, the proof of Theorem 1 extends to the case x = Oσn ‘. In such a large deviation range, it is more convenient to consider Pr”n = aφn‘• with a > 0 (cf. [20]). The saddlepoint equation (8) u0 r‘φn‘ = m then simplifies to u0 r‘ = a. Theorem 5. Let m = aφn‘ > 0. Suppose that the solution r of the equation u0 s‘ = a exists and satisfies −ρ ≤ r ≤ ρ. Then Pr”n = m• satisfies (11) uniformly in m. In particular, if us‘ = es − 1, then the result (11) can be stated differently:  X e2k r‘ φm n‘ −φn‘+vr‘ 1+ e Pr”n = m• = m! mk 1≤k≤ν   1 ŽrŽ 1 +O + ; + p mν+1 κn φn‘ κn φn‘

for some polynomials e2k r‘ of r = log a. In this case, it is more convenient to work with the probability generating function of n and apply Selberg’s method (cf. [29], [30], Chapter II.6), which is a variant of the usual saddlepoint method; see also [16, 17] for details. 7. Conclusion. The model that we developed in [16, 18, 19] and in this paper may be termed an “analytic scheme for moment generating functions” by which the similarity of the statistical properties of many apparently different structures (like the number of cycles in permutations and the number of prime factors in integers) is well explained by the analytic properties of

180

H.-K. HWANG

their moment generating functions. At first glance, these properties seem difficult to establish. But for concrete combinatorial and arithmetic problems, we have demonstrated, by using analytic methods, that analytic properties of moment generating functions are well reflected by the singularity type of the associated bivariate generating functions. Thus a classification according to the latter and then the use of suitable analytic methods, such as singularity analysis (cf. [10]), allow us to derive the required properties for moment generating functions in a rather systematic and general way. This explains roughly why there are so many similarities between the number of cycles in permutations and the number of connected components in 2-regular graphs [5], because the dominant singularity of the corresponding bivariate generating functions are both of type exp–log. Such an approach is also rather robust under structural perturbations when one considers, for example, structures without components of prescribed sizes or with some components appearing at most a specified number of times, and so on. A detailed study in this direction can be found in [16]. The uniformity afforded by the singularity analysis is also useful for other probabilistic properties of combinatorial parameters; see, for example, [1] and [15]. REFERENCES ´ S. (1995). Total variation asymptotics for Poisson pro[1] Arratia, R., Stark, D. and Tavare, cess approximations of logarithmic combinatorial assemblies. Ann. Probab. 23 1347– 1388. [2] Bender, E. A. (1973). Central and local limit theorems applied to asymptotic enumeration. J. Combin. Theory Ser. A 15 91–111. [3] Canfield, E. R. (1977). Central and local limit theorems for the coefficients of polynomials of binomial type. J. Combin. Theory Ser. A 23 275–290. [4] Chaganty, N. R. and Sethuraman, J. (1985). Large deviation local limit theorems for arbitrary sequences of random variables. Ann. Probab. 13 97–114. [5] Comtet, L. (1974). Advanced Combinatorics, the Art of Finite and Infinite Expansions, rev. ed. Reidel, Dordrecht. ´ H. (1970). Random Variables and Probability Distributions, 3rd ed. Cambridge [6] Cramer, Univ. Press. [7] Drmota, M. (1994). A bivariate asymptotic expansion of coefficients of powers of generating functions. European J. Combin. 15 139–152. ˝ P. (1948). On the integers having exactly k prime factors. Ann. Math. 49 53–66. [8] Erdos, [9] Flajolet, P. and Odlyzko, A. M. (1990). Random mapping statistics. Lecture Notes in Comput. Sci. 434 329–354. Springer, Berlin. [10] Flajolet, P. and Odlyzko, A. M. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216–240. [11] 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. [12] Flajolet, P. and Soria, M. (1993). General combinatorial schemes, Gaussian limit distributions and exponential tails. Discrete Math. 114 159–180. [13] Gao, Z. and Richmond, L. B. (1992). Central and local limit theorems applied to asymptotic enumeration IV, multivariate generating functions. J. Comput. Appl. Math. 41 177–186. [14] Greene, D. H. and Knuth, D. E. (1990). Mathematics for the Analysis of Algorithms, 3rd ¨ ed. Birkhauser, Boston. [15] Hansen, J. C. (1994). Order statistics for decomposable combinatorial structures. Random Structures Algorithms 5 517–533.

LOCAL LIMIT THEOREMS

181

[16] Hwang, H.-K. (1994). Th´eor`emes limites pour les structures combinatoires et les fonctions ´ arithm´etiques. Ph.D dissertation, Ecole Polytechnique, Palaiseau. [17] Hwang, H.-K. (1995). Asymptotic expansions for Stirling’s number of the first kind. J. Combin. Theory Ser. A 71 343–351. [18] Hwang, H.-K. (1996). Large deviations of combinatorial distributions I. Central limit theorems. Ann. Appl. Probab. 6 297–319. [19] Hwang, H.-K. (1996). Convergence rates in the central limit theorems for combinatorial structures. European J. Combin. To appear. [20] Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Science Publishers. [21] Kolchin, V. F. (1986). Random Mappings. Optimization Software, New York. [22] Kubilius, J. (1964). Probabilistic Methods in the Theory of Numbers. Amer. Math. Soc., Providence, RI. [23] Ledoux, M. (1992). Sur les d´eviations mod´er´ees des sommes de variables al´eatoires vectorielles ind´ependantes de mˆeme loi. Ann. Inst. H. Poincar´e Probab. Statist. 28 267–280. [24] Meir, A. and Moon, J. W. (1990). The asymptotic behaviour of coefficients of powers of certain generating functions. European J. Combin. 11 581–587. [25] Norton, K. K. (1978). Estimates for partial sums of the exponential series. J. Math. Anal. Appl. 63 265–296. [26] Otter, R. (1948). The number of trees. Ann. Math. 49 583–599. [27] Pavlov, A. I. (1988). Local limit theorems for the number of components of random permutations and mappings. Theory Probab. Appl. 33 183–187. [28] Richter, W. (1957). Local limit theorems for large deviations. Theory Probab. Appl. 2 206– 220. [29] Selberg, A. (1954). Note on a paper by L. G. Sathe. J. Indian Math. Soc. 18 83–87. [30] Tenenbaum, G. (1990). Introduction a` la Th´eorie Analytique et Probabiliste des Nombres. Institut Elie Cartan, Universit´e de Nancy I. Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei, 115 Taiwan E-mail: [email protected]