## Large deviations of combinatorial distributions II: Local limit theorems (1997)

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Citations: | 32 - 5 self |

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@MISC{Hwang97largedeviations,

author = {Hsien-Kuei Hwang},

title = {Large deviations of combinatorial distributions II: Local limit theorems},

year = {1997}

}

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### Abstract

This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables n#1 each of maximal span 1 (see below for definition), we are interested in the asymptotic behavior of the probabilities n = m} (m N, m = n x n # n , n := n , # n := n ), ##, where x n can tend to with n at a rate that is restricted to O(# n ). Our interest here is not to derive asymptotic expression for n = m} valid for the widest possible range of m, but to show that for m lying in the interval n O(# n ), very precise asymptotic formulae can be obtained. These formulae are in close connection with our results in [17]. 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 b # and h # > h such that X takes only values of the form b # + h # k

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Citation Context ... case of sums of independent and identically distributed random variables. His powerful analytic method which is then widely used consists of two major steps: the technique of associated distribution =-=[10] (wh-=-ose effect being to “shift” the mean) and the implicit use of the saddle-point method, see the next section for more details. His results together with his method have since been generalized in ma... |

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Citation Context ...n ≥ y} = Mn(s) e y −sw d � Mn(w) ≤ Mn(s)e −ys , for y > E Ωn, where s is chosen to satisfy the equation (13) or the saddle-point of the right side. This estimate is often referred as Chern=-=off’s bound [6]-=-, see also [34]. We remark that this technique frequently proves useful in different contexts under different names: for power series, this is known as the saddle-point estimate; and for Dirichlet ser... |

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Citation Context ... Q(w, ζ) 1 − z �−aw � � �� 1 − 1 1 + O log 2 (z ∼ ζ, z �∈ [ζ, ∞)), ζ 1 − z/ζ and P (w, z) is analytically continuable to a ∆-region. We can then apply the singularity =-=analysis of Flajolet and Odlyzko [12] to deduce the asymp-=-totic formula Pn(w) := [z n ]P (w, z) = ζ−n n aw−1 Γ(aw) eKw Q(w, ζ) � � 1 + O (log n) the O-term being uniform with respect to w, |w| ≤ η. For details, see [16]. 1 − 2 �� , (25) S... |

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Citation Context ...ldren’s yards, etc. [15]; 4. nodes of given out-degree in a random increasing tree in the polynomial variety [2]; 5. nodes of given out-degree in a random tree in the simply generated family of tree=-=s [30, 31]; 6.-=- “factorisatio numerorum” in (additive) arithmetical semigroups under Axiom A # [23]; 7. branching compositions of integers introduced in [18, Ch. 8]. Example 4. Arithmetical functions. Let fn,k d... |

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Citation Context ... have the same sign. Now log Mn(s) − su ′ (s)φ(n) = u(s)φ(n) + v(s) − su ′ (s)φ(n) + O( 1 ) = (u(s) − su ′ (s))φ(n) + O(s + 1 ), since v(0) = 0 and v(s) = O(s). Again, by Lagrange’s =-=inversion formula [39], we expand the function u(s) − su �-=-�� (s) in powers of ξ: where qm is given by It follows that with this choice of s as n → ∞. u(s) − su ′ (s) = − u2 2 ξ2 + � m≥3 qm = −1 m [wm−2 ]u ′′ � u ′ �−m (w) − ... |

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Citation Context ...e same algebraic and analytic schemes. This fact allows then a systematic treatment for their limiting properties and we first state the general conditions under which we are developing our arguments =-=[20, 19]. Let us assume that the moment ge-=-nerating functions Mn(s) of Ωn satisfy, as n → ∞, Mn(s) = E e Ωns � ∞ = e −∞ sy � � φ(n)u(s)+v(s) dWn(y) = e 1 + O κ −1 �� n , (4) uniformly for |s| ≤ ρ, s ∈ C, ρ > 0... |

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Citation Context ...r un-successful) searches in a random binary search tree [29, §2.4 and 2.5], a = 2; 5. The depth of a random node in a random increasing tree in a polynomial variety of degree d, d ≥ 2, a = d/(d ��=-=� 1) [2]; 6.-=- “Prime-divisor” functions in (additive) arithmetical semigroups under Axiom A # [24]. Example 3. Algebraico-logarithmic scheme. Let G(z) be the generating function of a subset of positive integer... |

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Citation Context ... a sufficiently small constant. Consequently, there exist two constants C1, C2 > 0 such that |λn(t)| ≤ exp � − t2 2 + C1|t| κn + C2(|t| + |t| 3 � ) � φ(n) (|t| ≤ Tn). We now apply Essee=-=n’s inequality [9, 33]-=- to establish (12). Thus the estimate of the difference of two distribution functions is reduced to the corresponding problem for associated characteristic functions. It is sufficient to show that Jn ... |

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Citation Context ...= M n (s) Z 1 y e sw d f M n (w) M n (s)e ys ; for y > E n , where s is chosen to satisfy the equation (13) or the saddle-point of the right side. This estimate is often referred as Cherno's bound [6], see also [34]. We remark that this technique frequently proves useful in dierent contexts under dierent names: for power series, this is known as the saddle-point estimate; and for Dirichlet serie... |

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Citation Context ...eorem 1 in the next section. Many immediate consequences of this result will be given in section 3. We then apply formulae (5) and (6) to the combinatorial distributions studied by Flajolet and Soria =-=[15, 16]-=- in section 4. Finally, we shall briefly discuss some examples from many different applications. Throughout this paper, all generating functions (ordinary, exponential, probability, characteristic fun... |

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Citation Context ...n for more details. His results together with his method have since been generalized in many directions by many authors, for detailed information, applications, and references, consult the monographs =-=[33, 35, 3, 37, 25]. -=-1.1 Main result Let {Ωn}n be a sequence of random variables with distribution functions Wn(x). A number of such sequences related to combinatorial structures and arithmetical functions have moment g... |

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Citation Context ...enerating functions, the source of which being not specified. Before we enter into details, let us briefly mention some relevant results in the probability literature. Cramér, in his pioneering paper=-= [7]-=-, first establishes general theorems for probabilities of large deviations in the case of sums of independent and identically distributed random variables. His powerful analytic method which is then w... |

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Citation Context ...limit distribution, local behaviour, etc.) of parameters in a large class of combinatorial structures are well reflected by the (dominant) singularity type of the associated generating functions, see =-=[4, 5, 15, 16, 17]-=-. The classification according to the latter leads to the study of (analytic) combinatorial schemes on which important progress has recently been made by Flajolet and Soria [15, 16]. In their papers, ... |

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Citation Context ...lynomials, cf. [38]. Our theorems apply to all these polynomials which also appear in the combinatorial study of histories of a number of data structures: stacks, priority queues, dictionaries, etc., =-=[11]-=-. In the other direction, probabilistic methods have been used to derive classical asymptotic expansions of certain orthogonal polynomials, cf. [28]. 6 Concluding remarks We have hitherto applied our ... |

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Citation Context ...em 1 applies with u(s) = − log(s/(e s − 1)) and φ(n) = n + 1. Local limit theorems for Ωn can be found in [1]. Another example of the same class is the level number sequence of Flajolet and Pro=-=dinger [13]. There -=-they consider the number hn of level number sequences for binary trees which is equivalent to the cardinality of the set Hn = � k Hnk, where Hnk = {(n1, n2, . . . , nk) : n1 = 1; 1 ≤ nj ≤ 2nj−... |

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Citation Context ...|z| < 1), We list some examples belonging to the same class the description of which can be found in the cited reference. Many other examples to which Theorem 3 applies can be found in [36, §6.2.6.3]=-= [24]-=- [18, Ch. 5]. 1. The number of connected components in a random mapping, a = 1 2 [15]; 2. The number of monic irreducible factors in a random monic polynomial of Fq[z], a finite field with q elements,... |

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Citation Context ...fficients of classical polynomials (not necessarily orthogonal), like Laguerre, Gegenbauer (which includes Chebyshev and Legendre), Charlier, Meixner (first kind), Lerch, and Humbert polynomials, cf. =-=[38]-=-. Our theorems apply to all these polynomials which also appear in the combinatorial study of histories of a number of data structures: stacks, priority queues, dictionaries, etc., [11]. In the other ... |

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Citation Context ...C is supposed to satisfy ζ < 1. A result parallel to Theorem 3 can be derived by replacing a log n with a log log n. This will have important applications to some additive arithmetical functions, cf.=-= [26]. 4.2 Algebraico-loga-=-rithmic scheme Next, let us consider generating functions of the form P (w, z) = � n,m≥0 pnmw m z n = 1 (1 − wG(z)) α � �β 1 log 1 − wG(z) 13 (G(0) = 0),swhere β ∈ N, α ≥ 0, and α... |

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Citation Context ...e the zone of normal convergence contains the interval 0 ≤ x ≤ X(n), where X(n) = o(n 1 3 ). Other examples include: 1. the number of blocks in a random ordered set-partition and cyclic set-partit=-=ion [16, 17]: 1 1 − w(e-=- z − 1) and log 1 1 − w(e z − 1) ; 2. the number of connected components (or cycles) in a random alignment [14, §2.3] and its cyclic counterpart: 1 1 + w log(1 − z) and log 1 1 + w log(1 − ... |

9 | The asymptotic behaviour of coefficients of powers of certain generating functions, submitted for publication - Meir, Moon |

8 |
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Citation Context ...lynomial variety [2]; 5. nodes of given out-degree in a random tree in the simply generated family of trees [30, 31]; 6. “factorisatio numerorum” in (additive) arithmetical semigroups under Axiom =-=A # [23]; -=-7. branching compositions of integers introduced in [18, Ch. 8]. Example 4. Arithmetical functions. Let fn,k denote the number of factorizations of n into k integer factors greater than 1, n ≥ 2, k ... |

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Citation Context ...es: for power series, this is known as the saddle-point estimate; and for Dirichlet series, this is known as Rankin’s technique. All these can be formulated in the form of Laplace-Stieltjes transfor=-=m [32]. 3 -=-Some corollaries of Theorem 1 From the results established in the previous section, namely, formulae (5) and (6), we can deduce some interesting corollaries as in [7, Thms 2–4][33, pp. 228–230]. F... |

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7 |
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Citation Context ...lynomial variety [2]; 5. nodes of given out-degree in a random tree in the simply generated family of trees [30, 31]; 6. \factorisatio numerorum" in (additive) arithmetical semigroups under Axiom=-= A # [23-=-]; 7. branching compositions of integers introduced in [18, Ch. 8]. Example 4. Arithmetical functions. Let f n;k denote the number of factorizations of n into k integer factors greater than 1, n 2, k... |

6 |
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Citation Context ...tures: stacks, priority queues, dictionaries, etc., [11]. In the other direction, probabilistic methods have been used to derive classical asymptotic expansions of certain orthogonal polynomials, cf. =-=[28]-=-. 6 Concluding remarks We have hitherto applied our Theorem 1 to discrete random variables. It might be possible that it finds application in other fields. It seems of some interest to replace the err... |

6 | Méthode d’Analyse pour les Constructions Combinatoires et les Algorithmes. Thèse d’ État, Université de Paris-Sud - Soria - 1990 |

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3 |
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Citation Context ...ldren’s yards, etc. [15]; 4. nodes of given out-degree in a random increasing tree in the polynomial variety [2]; 5. nodes of given out-degree in a random tree in the simply generated family of tree=-=s [30, 31]; 6.-=- “factorisatio numerorum” in (additive) arithmetical semigroups under Axiom A # [23]; 7. branching compositions of integers introduced in [18, Ch. 8]. Example 4. Arithmetical functions. Let fn,k d... |

3 | Large deviation local limit theorems for arbitrary sequences of random variables - Chaganty, Sethuraman - 1985 |