Local Limit Theorem for the Multiple Power Series Distributions

: We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufﬁcient conditions for the integral limit theorem were obtained. Here, the necessary and sufﬁcient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.


Introduction
Let (a(i) ≥ 0, i = 0, 1, 2, . . . ) be a sequence with 1). The trivial case a(i) ≡ 0, i = 0, 1, 2, . . . is excluded. It is said that a random variable ξ x has a power series distribution iff for some B(x) and for any i ∈ Z + . Power series distributions were introduced in the fundamental paper of Noack [1] (1950). Systematic studies of their properties (moments, generating functions, convolutions, limit properties, statistical applications, etc.) began immediately. References may be found in the encyclopedias of Johnson, Kotz, and Kemp [2] (for the one-dimensional case) and Johnson, Kotz, and Balakrishnan [3] (for the multidimensional case). For example, the binomial, Poisson, negative binomial, and logarithmic distributions, as well as their multidimensional analogues are among the important distributions in this class.
Note that power series distributions are widely useful in a generalized allocation scheme (in the one-dimensional case). This scheme was introduced by V. Kolchin [4]. His results and, in particular, those obtained with the use of this scheme, play an important role in probabilistic combinatorics (see, for example, his books [5,6]). So, one can express distributions of various characteristics of random permutations (a(i) = 1/i), random mappings ((a(i) = i −1 ∑ i−1 k=0 i k /k!)) [5]), and random mappings with various constraints (on cycle length, height, component sizes, etc.; see, for example, the books of Timashev [7,8]), random trees, and random forests (i.e., random mappings with cycles of only unit length (see the book of Yu. Pavlov [9])) in terms of power series distributions. An analogue of Kolchin's generalized allocation scheme [4] with a bounded number of particles was introduced in the work of A.N. Chuprunov and I. Fezekash [10]. A corresponding multivariate scheme was recently introduced by A.N. Chuprunov, G. Alsaied, and M. Alkhuzani [11]. For another investigation of A.N. Chuprunov and his students, see the paper [11] and the references therein. We also note the successful work of the representatives of the Karelian Scientific Center in the study of the asymptotic properties of configuration graphs under the leadership and participation of Yu.L. Pavlov by I.A. Cheplyukova, M.M. Leri, and E.V. Khvorostyanskaya [12][13][14][15][16][17].
Suppose that B(x) regularly varies as x ↑ 1 with index > 0 [18,19]. It is known, in this case, that In addition, the corresponding local limit theorem is true when a(i) is regularly varying at infinity with index − 1 > −1. See, for instance, Timashev [8].
The multidimensional integral limit theorem was obtained in [20]. It is supposed in [20] that the corresponding multiple power series regularly varies at the boundary point of its convergence (see Definition 2). In [21], it was shown that this condition is necessary and sufficient.
In this paper, we prove the corresponding local limit theorem. For this aim, we introduce in Section 2 some generalizations of multivariate regularly varying sequences in the orthant. Namely, the notion of R-weakly one-sided oscillatory sequences at infinity along some sequence (see Definition 3). This concept allows us to give adequate conditions for the validity of both the local limit theorem and the corresponding statement of Tauberian type (Lemma 2). The definition of multiple power series distribution and the main result are given in the next section (see Definition 1 and Theorem 1, respectively). Here, we also formulate the corresponding integral limit result from [21] as Lemma 1. The statement of this lemma also gives the necessary and sufficient conditions but describes them in terms of regular variation of the power series B(x) at the boundary point of their existence. Proofs of Lemma 2 and the main result (Theorem 1) are given in the Sections 3 and 4, respectively. In Section 5, we describe some previous results in this direction.

Multiple Power Series Distributions
First we give the necessary definitions. Let (a(i) ≥ 0, i ∈ Z n + ) be a multiple sequence with ) ξ x has a multiple power series distribution iff for some B(x) and for any i ∈ Z n + .
It is clear that P{ξ x ∈ R n } = 1. The history of this notion and some bibliographic references are given in encyclopedias [2,3], also see the articles [20,22]. Let the sequence of vectors for an arbitrary fixed λ = (λ 1 , . . . , λ n ) > 0 as k → ∞.
The following statement has been proved in [21] (we formulate it as a lemma).
Let R(k) be some positive sequence. To formulate the resulting limit theorem, we need to give the following definition.
Hereinafter, we define a(x) = a([x]) for x / ∈ Z n + . The simplest examples of such sequences are monotone in each variable sequence (a(i) ≥ 0, i ∈ Z n + ).

Theorem 1.
Suppose that B(x) regularly varies as x ↑ 1 along the sequence b = b(k) (i.e., the the assumption of integral limit Lemma 1 is true). Then, for any compact K ⊂ G and for any (some) fixed vector u ∈ G and x = exp(−u/b) where function ψ u (·) is continuous in G, iff the sequence a(i) is R-weakly one-sided oscillatory at infinity along the sequence b = b(k) with In both cases, the measure Φ(·) from Lemma 1 has the continuous density ϕ(·) in G and the following equality holds: , ∀y ∈ G.

Tauberian Lemma
The next lemma gives some generalization of the Tauberian Theorem 2 from [23].

Lemma 2.
Assume that B(x) regularly varies as x ↑ 1 along the sequence b = b(k) (i.e., (2.2) holds). Then, for some continuous function ϕ(·) in G the relation holds for any compact K ⊂ G iff the sequence a(i) is R-weakly one-sided oscillatory at infinity along the sequence b = b(k) with R(k) from Equation (7). In both cases, the measure Φ(·) from Lemma 1 is absolutely continuous in G with density ϕ(·).

On Some Previous Results
The definition of regularly varying functions of one variable was given in Karamata's well-known work [25]. The notion of regularly varying functions at infinity along some sequence in an orthant was introduced in Omey [26]. The definition of regularly varying multiple power series is given in [23]. A brief overview of various definitions of multivariate regularly varying functions is available in [27]. The history of different class functions having slow (one-sided or ordinary) oscillation can be seen in the book [24]. In [22], we give the integral representation and Abelian statements (Theorems 3.1 and 3.2). With the help of these theorems, it is easy to set such sequences a(i) explicitly.
As the source, for n = 1 the sufficient condition for Equation (2.3) was given in Timashev [28], see also [8]. In [20], we show that conditions from [8,28] are equivalent to Equation (2.2). Timashev used the method of moments in his aforementioned result. In the papers [20,22,27] and in this article, we use the corresponding Tauberian statements. All these statements go back to Karamata's well-known Tauberian theorems [29,30].