**1. Introduction**

Let (*a*(*i*) ≥ 0, *i* = 0, 1, 2, . . .) be a sequence with

$$B(\mathbf{x}) = \sum\_{i=0}^{\infty} a(i)\mathbf{x}^i < \infty$$

for *x* ∈ (0, <sup>1</sup>). The trivial case *a*(*i*) ≡ 0, *i* = 0, 1, 2, ... is excluded. It is said that a random variable *ξx* has a power series distribution iff

$$\mathbb{P}\{\mathfrak{f}\_{\mathbf{x}}=i\} = \frac{a(i)\mathbf{x}^{i}}{B(\mathbf{x})},$$

for some *<sup>B</sup>*(*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 ∑*<sup>i</sup>*−<sup>1</sup> *k*=0 *i <sup>k</sup>*/*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 [12–17].

Suppose that *<sup>B</sup>*(*x*) regularly varies as *x* ↑ 1 with index > 0 [18,19]. It is known, in this case, that

$$\mathcal{P}\{\xi\_x(1-x) \le y\} \to \int\_0^y e^{-u} u^{\varrho -1} \, du, \,\,\forall y \ge 0$$

as *x* ↑ 1. 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 *<sup>B</sup>*(*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.

## **2. Main Result**

## *2.1. Some Notations*

We introduce the following notations. Let the vectors *x* = (*<sup>x</sup>*1, ... , *xn*) and *y* = (*y*1, ... , *yn*) belong to *Rn*. Denote *xy* = (*<sup>x</sup>*1*y*1, ... , *xnyn*) and *x*/*y* = (*x*1/*y*1, ... , *xn*/*yn*) (the last in the case, when *yk* -= 0 ∀*k* = 1, . . . , *n*). Put exp(*x*)=(exp(*<sup>x</sup>*1), ... , exp(*xn*)), ln *x* = (ln *x*1, ... , ln *xn*). The notation *x* ↑ **1** means that *x* → **1**, *x* ∈ (0, <sup>1</sup>)*<sup>n</sup>*. Here **1** = (1, ... , 1). Set *Rn* + = {*x* : *x* = (*<sup>x</sup>*1, ... , *xn*) ∈ *Rn*, *xk* ≥ 0 ∀*k* = 1, ... , *<sup>n</sup>*}, *Zn* + = {*x* : *x* = (*<sup>x</sup>*1, ... , *xn*) ∈ *Rn* +, *xk* ∈ *Z*+ = *N* ∪ 0 ∀*k* = 1, ... , *<sup>n</sup>*}. For *α* = (*<sup>α</sup>*1,..., *<sup>α</sup>n*) ∈ *Rn* +, *x* = (*<sup>x</sup>*1,..., *xn*) ∈ *Rn* + we use an abbreviation

$$\mathbf{x}^{a} = \prod\_{k=1}^{n} \mathbf{x}\_{k}^{a\_{k}}.$$

assuming that 00 = 1. Let (*ηk*, *k* ∈ *N*) be a sequence of random vectors (r.v.) from *Rn*. Further, the notation *ηk d* → *η* means the weak convergence of the corresponding distributions with <sup>P</sup>{*η* ∈ *Rn*} = 1.

#### *2.2. Multiple Power Series Distributions*

First we give the necessary definitions. Let (*a*(*i*) ≥ 0, *i* ∈ *Zn* +) be a multiple sequence with

$$B(\mathbf{x}) = \sum\_{i \in Z\_{\pm}^n} a(i)x^i < \infty$$

for *x* = (*<sup>x</sup>*1,..., *xn*) ∈ [0, <sup>1</sup>)*<sup>n</sup>*. The trivial case *a*(*i*) ≡ 0, *i* ∈ *Zn* + is excluded. **Definition 1.** *For x* ∈ [0, 1)*n a random vector (r.v.) ξx has a multiple power series distribution iff*

$$\mathbb{P}\{\mathfrak{f}\_x = i\} = \frac{a(i)x^i}{B(x)},\tag{1}$$

*for some <sup>B</sup>*(*x*) *and for any i* ∈ *Zn*+*.*

It is clear that <sup>P</sup>{*ξx* ∈ *Rn*} = 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 *b* = *b*(*k*) ∈ (0, <sup>∞</sup>)*<sup>n</sup>*, *k* ∈ *N* be given with *bj* = *bj*(*k*) → <sup>∞</sup>, ∀*j* = 1, . . . , *n* as *k* → ∞.

**Definition 2** ([23])**.** *We say that <sup>B</sup>*(*x*) *regularly varies as x* ↑ **1** *along the sequence b* = *b*(*k*)*, iff*

$$\frac{B(\exp(-\lambda/b))}{B(\exp(-\mathbf{1}/b))} \to \Psi(\lambda) \in (0, \infty),\tag{2}$$

*for an arbitrary fixed λ* = (*<sup>λ</sup>*1,..., *<sup>λ</sup>n*) > 0 *as k* → ∞*.*

(Notations *λ*/*b* and exp(−*λ*/*b*) are defined in the Section 2.1). The following statement has been proved in [21] (we formulate it as a lemma).

**Lemma 1.** *A series <sup>B</sup>*(*x*) *regularly varies as x* ↑ **1** *along the sequence b* = *b*(*k*) *iff for any (some) fixed vector u* ∈ *G and x* = exp(−*<sup>u</sup>*/*b*)

$$\prescript{d}{}{\zeta\_x}(\mathbf{1}-\mathbf{x}) \xrightarrow{d} \eta = \eta(\mathsf{u}), \quad (k \to \infty) \tag{3}$$

*In both cases, the function* Ψ(*λ*) *from Equation (2) is the Laplace transform of some σ-finite measure* <sup>Φ</sup>(·) *and r.v. η*(*u*) *has Laplace transform* Ψ((*λ* + **<sup>1</sup>**)*u*)/Ψ(*u*)*.*

Let *R*(*k*) be some positive sequence. To formulate the resulting limit theorem, we need to give the following definition.

**Definition 3.** *We say that the sequence a*(*i*) *is R-weakly one-sided oscillatory at infinity along the sequence b* = *b*(*k*) *if for every j* = 1, ... , *n and for any sequence zj* = *zj*(*k*) > 1, *zj* = 1 + *o*(1) *one of the following inequalities*

$$\liminf\_{k \to \infty} (a(r\_1, \dots, r\_{j-1}, z\_j r\_j, r\_{j+1}, \dots, r\_n) - a(r)) / \mathcal{R}(k) \ge 0; \tag{4}$$

$$\limsup\_{k \to \infty} (a(r\_1, \dots, r\_{j-1}, z\_j r\_{j'}, r\_{j+1}, \dots, r\_n) - a(r)) / R(k) \le 0. \tag{5}$$

*holds for every fixed y* = (*y*1, ... , *yn*) ∈ *G. Here r* = *<sup>r</sup>*(*k*)=(*<sup>r</sup>*1(*k*), ... ,*rn*(*k*)) *is an arbitrary function of k with*

$$r\_1 \sim y\_1 b\_1, \dots, r\_n \sim y\_n b\_n.$$

Hereinafter, we define *a*(*x*) = *a*([*x*]) for *x* ∈/ *Zn*+. The simplest examples of such sequences are monotone in each variable sequence (*a*(*i*) ≥ 0, *i* ∈ *Zn*+).

**Theorem 1.** *Suppose that <sup>B</sup>*(*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(−*<sup>u</sup>*/*b*)

$$\frac{\mathbb{P}\{\xi\_{\mathbf{x}} = [y/(\mathbf{1} - \mathbf{x})]\}}{\prod\_{j=1}^{n} (1 - x\_{j})} \stackrel{y \in K}{\Longrightarrow} \psi\_{u}(y) < \infty \quad (k \to \infty) \tag{6}$$

*where function ψu*(·) *is continuous in G, iff the sequence a*(*i*) *is R-weakly one-sided oscillatory at infinity along the sequence b* = *b*(*k*) *with*

$$R(k) = B(\exp(-\mathbf{1}/b(k))) / \prod\_{i=1} b\_i(k). \tag{7}$$

*In both cases, the measure* <sup>Φ</sup>(·) *from Lemma 1 has the continuous density ϕ*(·) *in G and the following equality holds:*

$$\psi\_u(y) = \frac{\varrho(y/u)e^{-(y,\mathbf{1})}}{\prod\_{j=1}^n u\_j \Psi(u)}, \quad \forall y \in \mathcal{G}. \tag{8}$$

Note that, in Theorem 1, the case when Φ(*∂G*) > <sup>0</sup>(⇔ <sup>P</sup>{*η*(*u*) ∈ *∂G*} > 0) is not excluded. In addition, we admit that *ψu*(*y*) = 0, *y* ∈ *V*, for some nonempty set *V* ⊆ *G* in this theorem.

## **3. Tauberian Lemma**

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

**Lemma 2.** *Assume that <sup>B</sup>*(*x*) *regularly varies as x* ↑ **1** *along the sequence b* = *b*(*k*) *(i.e., (2) holds). Then, for some continuous function ϕ*(·) *in G the relation*

$$\frac{a(b\upsilon)}{R(k)} \stackrel{\upsilon \in K}{\implies} \varrho(\iota) < \infty \tag{9}$$

*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* <sup>Φ</sup>(·) *from Lemma 1 is absolutely continuous in G with density ϕ*(·)*.*

**Proof.** For an arbitrary bounded set *A* ⊂ *Rn*+, put

$$\Phi\_k(A) = \sum\_{i \in \mathbb{Z}\_{+}^n, \ i/b \in A} \frac{a(i)}{\prod\_{j=1}^n m\_j(k)\mathcal{R}(k)} \tag{10}$$

It follows from Equations (2) and (7) that

$$\begin{aligned} \check{\Phi}\_k(y) &\equiv \int\_{R\_+^n} e^{-(x,y)} \Phi\_k(dx) = \frac{B(\exp(-y/b))}{\prod\_{j=1}^n b\_j(k)R(k)} \\ &\rightarrow \psi(y) = \check{\Phi}(y) \equiv \int\_{R\_+^n} e^{-(x,y)} \Phi(dx) \end{aligned}$$

for any fixed *y* ∈ *G*. The last equality follows from the statement of Lemma 1. Thus, according to the continuity theorem for Laplace transforms of measures, it follows from Equation (10) that

$$
\Phi\_k(\cdot) \Rightarrow \Phi(\cdot). \tag{11}
$$

(see, for example the theorem 1.3.2 from [24]). Suppose that the sequence *a*(*i*) is *R*-weakly one-sided oscillatory at infinity along the sequence *b* = *b*(*k*). Set *m*(*j*) = 1 if Equation (4) holds and *m*(*j*) = −1 if Equation (5) is valid. Fix *v* ∈ *G*. For an arbitrary *δ* ∈ (0, <sup>1</sup>), put

$$A(\delta) = \left\{ y = (y\_1, \dots, y\_n), \ y\_j \in \left( v\_j, v\_j (1 + \delta)^{b(j)} \right), \ \forall j = 1, \dots, n \right\} \tag{12}$$

(for *c* > *d*, we put (*c*, *d*)=(*d*, *c*)). Further, for an arbitrary *ε* ∈ (0, <sup>1</sup>), there exists such *δ* ∈ (0,*ε*) that

$$\frac{a(i) - a(bv)}{R(k)} \ge -\varepsilon \tag{13}$$

for any *i* ∈ *mA*(*δ*). The proof of this fact repeats the proof of Lemma 5 from [23]. Without loss of generality, we assume that Φ(*∂A*(*δ*)) = 0. It follows from Equations (9) and (13) that

$$\Phi\_k(A(\delta)) = \sum\_{\substack{i \in Z\_+^n, \ i/b \in A(\delta)}} \frac{a(i)}{\prod\_{j=1}^n b\_j(k)R(k)}$$

$$\ge -\varepsilon + \frac{a(b\upsilon)}{R(k)} \frac{1}{\prod\_{j=1}^n b\_j(k)} \sum\_{\substack{i \in Z\_+^n, \ i/b \in A(\delta)}} 1.$$

$$\ge -\varepsilon + \frac{a(b\upsilon)}{R(k)} (1 + \eta\_k) |A(\delta)|$$

where *ηk* → 0 as *k* → ∞. By |*A*(*δ*)|, we denote here the Lebesque measure of the set *<sup>A</sup>*(*δ*). Therefore,

$$\frac{a(bv)}{R(k)} \le \left(\frac{\Phi\_k(A(\delta))}{|A(\delta)|} + \frac{\varepsilon}{|A(\delta)|}\right) \frac{1}{1 + \eta\_k}.\tag{14}$$

Since Φ(*∂A*(*δ*)) = 0, we have from Equations (9) and (11) that

$$\Phi\_k(A(\delta)) = \sum\_{\substack{i \in \mathbb{Z}\_+^n, \ i/b \in A(\delta)}} \frac{a(i)}{\prod\_{j=1}^n b\_j(k)\mathbb{R}(k)} \to \Phi(A(\delta)).\tag{15}$$

Tending in Equation (14) *k* to ∞ and using Equation (15), we have

$$\limsup\_{k \to \infty} \frac{a(bv)}{R(k)} \le \frac{\Phi(A(\delta))}{|A(\delta)|} + \frac{\varepsilon}{|A(\delta)|}. \tag{16}$$

Since the left side of Equation (16) does not depend on *ε*, we have

$$\limsup\_{k \to \infty} \frac{a(bv)}{R(k)} \le \frac{\Phi(A(\delta))}{|A(\delta)|}. \tag{17}$$

Put Δ = { *δ* ∈ (0, 1) : Φ(*∂A*(*δ*)) = <sup>0</sup>}. Since the left side of Equation (17) does not depend on *δ*, we have

$$\limsup\_{k \to \infty} \frac{a(bv)}{R(k)} \le \liminf\_{\delta \to 0, \ \delta \in \Lambda} \frac{\Phi(A(\delta))}{|A(\delta)|}. \tag{18}$$

Similarly, we obtain the inequality

$$\liminf\_{k \to \infty} \frac{a(bv)}{R(k)} \ge \limsup\_{\delta \to 0, \,\delta \in \Lambda} \frac{\Phi(A(\delta))}{|A(\delta)|}. \tag{19}$$

It follows from Equations (18) and (19) that there exist the next two limits:

$$\lim\_{k \to \infty} \frac{a(b\upsilon)}{\mathcal{R}(k)} = \lim\_{\delta \to 0, \,\delta \in \Lambda} \frac{\Phi(A(\delta))}{|A(\delta)|} \left( \stackrel{def}{=} \,\,\!\! \! \! \! \! / \,\!\! / \,\!\! / \,\!\! / \,\!\!\! / \,\!\!\! / \,\!\!\! / \,\!\!\!\/) . \,\!\!\!\!\/,\!$$

The next proof repeats the proof of Theorem 2 from [23]. The inverse assertion of Lemma 2 follows immediately from Equation (9). Lemma 2 is proved.

#### **4. Proof of Theorem 1**

Suppose that Equation (2) holds. Put for *z* ∈ *Nn* and *x* ∈ [0, 1]*n*

$$p(z,x) = \mathbb{P}\{\zeta\_x = z\} = \frac{a(z)}{B(x)} \exp(z, \ln x).$$

We have

$$a(z) = p(z, \mathbf{x}) B(\mathbf{x}) \exp(z, -\ln \mathbf{x}).\tag{20}$$

Suppose that Equation (6) takes place for some *u* ∈ *G* and continuous in *G* function *ψu*(·). For fixed *y* ∈ *G*, put in Equation (20) *x* = exp(−*<sup>u</sup>*/*b*(*k*)) and *z* = [*y*/(**<sup>1</sup>** − *<sup>x</sup>*)]. We have *x* = **1** − (*u* + *ε*(*k*))/*b*(*k*) and *z* = [*b*(*k*)*y*/(*u* + *ε*(*k*))] = *b*(*k*)(*y*/*u* + *<sup>δ</sup>*(*k*)). Here *ε*(*k*) and *δ*(*k*) are some functions tending to zero as *k* → ∞. Thus (*z*, − ln *<sup>x</sup>*)=(*y*, **1**) + *o*(1) as *k* → ∞. So, it follows from Equations (20), (2), and (6) that

$$a(z) = p(z, \mathbf{x})\mathbb{B}(\mathbf{x})\exp(z, -\ln \mathbf{x}) = (1 + o(1))p(z, \mathbf{x})\mathbb{B}(\mathbf{x})\exp(y, \mathbf{1})$$

$$= \prod\_{j=1}^{n} (1 - \mathbf{x}\_{j})\mathbb{B}(\mathbf{x})(\psi\_{u}(y) + o(1))\exp(y, \mathbf{1})$$

$$= \prod\_{j=1}^{n} (1 - \mathbf{x}\_{j})\mathbb{B}(\exp(-\mathbf{1}/b(k))\Psi(u)(\psi\_{u}(y) + o(1))\exp(y, \mathbf{1})$$

$$= \prod\_{j=1}^{n} (u\_{j}/b\_{j}(k))\mathbb{B}(\exp(-\mathbf{1}/b(k))\Psi(u)(\psi\_{u}(y) + o(1))\exp(y, \mathbf{1})$$

$$= R(k)\prod\_{j=1}^{n} u\_{j}\Psi(u)(\psi\_{u}(y) + o(1))\exp(y, \mathbf{1})\tag{21}$$

according to Equation (7). Since Equation (6) holds locally uniformly on *y* then it follows from Equation (21) that

$$\frac{a(by/u)}{R(k)} \to \prod\_{j=1}^{n} u\_j \Psi(u) \psi\_u(y) \exp(y/\mathbf{1}) = \wp(y/u) \tag{22}$$

and the last relation also holds locally uniformly on *y*. The equality Equation (8) follows directly from Equation (22). Replacing in Equation (22) *y*/*u* by *v*, we obtain Equation (9). One-sided *R*-oscillation of *<sup>a</sup>*(·) along *b*(*k*) follows immediately from Equation (9). The proof of inverse assertion repeats the proof of Theorem 2 from [20].

#### **5. 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 (3) was given in Timashev [28], see also [8]. In [20], we show that conditions from [8,28] are equivalent to Equation (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].

**Funding:** This research received no external funding.

**Acknowledgments:** The author expresses his deep gratitude to reviewers for valuable comments.

**Conflicts of Interest:** The author declare no conflict of interest.
