PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 80(94) (2006), 253–258 DOI:10.2298/PIM0694253S

ON REGULARLY VARYING MOMENTS FOR POWER SERIES DISTRIBUTIONS

Slavko Simić

Abstract. For the power series distribution, generated by an entire function of finite order, we obtain the asymptotic behavior of its regularly varying moments. Namely, we prove that E_{w}X^{α}ℓ(X) ∼ (E_{w}X)^{α}ℓ(E_{w}X), α > 0 (w → ∞), where ℓ(·) is an arbitrary slowly varying function.

0. Introduction

0.1. Denote by A_{ρ} the class of transcendental entire functions with positive Taylor coefficients and of finite order ρ, 0 ⩽ ρ < ∞.

Definition 1. Let f(w) = ∑ a_{n}w^{n}, f ∈ A_{ρ}. A power series distribution with parameter w > 0, generated by f, is defined by (cf. [2])

P (X = n) := a_{n}w^{n}/f(w), n = 0, 1, 2, ⋯

Our aim is to obtain the asymptotic behavior of the k-th moment E_{w}X^{k} when w → ∞, where

E_{w}X^{k} := ∑ n^{k}P (X = n) = ∑ n^{k}a_{n}w^{n}/f(w), k = 1, 2, ⋯

Note that the expectation E_{w}X is equal to

(1)  E_{w}X := ∑ na_{n}w^{n}/f(w) = wf′(w)/f(w).

For any k, consider the sequence of functions f_{k}(w) defined recursively by

f_{k}(w) = wf′_{k−1}(w), k = 1, 2, ⋯; f_{0}(w) = f(w) = ∑ a_{n}w^{n}.

Then f_{k}(w) = ∑ n^{k}a_{n}w^{n} ∈ A_{ρ} and

(2)  E_{w}X^{k} = f_{k}(w)/f(w), k = 1, 2, ⋯

We shall derive the asymptotic behavior of E_{w}X^{k} for large w by applying our following recent result:

2000 Mathematics Subject Classification: Primary 60E05; Secondary 30D15.

Key words and phrases: regular variation, moments, power series distributions.

Supported by the Ministry of Science of Serbia, Grant 144021.

Theorem 1. [5] For an arbitrary f ∈ A_{ρ}, we have

\frac{f(w)f′′(w)}{(f′(w))^{2}} → 1 (w → ∞).

independently of the order ρ.

0.2. Further generalization leads to the concept of regularly varying moments E_{w}X^{α}ℓ(X) (cf. [1, p. 335]),

(3)  E_{w}X^{α}ℓ(X) := ∑ n^{α}ℓ(n)a_{n}w^{n}/f(w),

where α is a positive real number and ℓ(·) is a slowly varying function.

Definition 2. A positive continuous function ℓ(·), defined on [x_{0}, ∞), is slowly varying if the asymptotic equivalence ℓ(tx) ∼ ℓ(x), (x → ∞), holds for each t > 0.

For x ∈ [0, x_{0}) we can take f(x) := f(x_{0}). Some examples of slowly varying functions are

log^{a} x; log^{b}(log x); exp(log^{c} x); exp(log x/ log log x); a, b ∈ R, 0 < c < 1.

Functions g(·) of the form g(x) = x^{μ}ℓ(x) are regularly varying with index μ ∈ R (cf. [1, p.18]). Each regularly varying function x^{μ}ℓ(x) generates a regularly varying sequence of the form {n^{μ}ℓ(n)}_{n=1}^{∞}.

The main tool for asymptotic estimation of regularly varying moments is the following theorem on matrix transforms with slowly varying sequences (cf. [4]).

Theorem 2. For a given complex-valued matrix (A_{nk})_{n,k=1}^{∞} define t_{n}(ρ) := ∑ k^{ρ}|A_{nk}|. Suppose that for some positive constants a, A, t_{n}(ρ) exists for −a ⩽ ρ ⩽ 1 and, for sufficiently large n,

(i) |∑ A_{nk}| ⩾ A; (ii) t_{n}(0) → 1; (iii) t_{n}(1) → ∞; (iv) t_{n}(−a) = O((t_{n}(1))^{−a}).

Then the asymptotic relation

∑ A_{nk}ℓ(k) = ℓ(t_{n}(1)) (∑ A_{nk}) (1 + o(1)) (n → ∞),

holds for all slowly varying sequences {ℓ(k)}_{k=1}^{∞}.

0.3. Here we quote some well-known assertions we shall need in the sequel.

Lemma 1. If a(x) ∼ b(x) → ∞, then ℓ(a(x)) ∼ ℓ(b(x)) (x → ∞).

Lemma 2. [3, Vol. I, p. 36]. Let g(x) = ∑ a_{n}x^{n}, h(x) = ∑ b_{n}x^{n}, g, h ∈ A_{ρ}. If a_{n} ∼ b_{n} (n → ∞), then g(x) ∼ h(x) (x → ∞).

Lemma 3. Jensen’s inequality: EX^{t} ⩾ (EX)^{t}, t > 1, and vice versa for 0 < t < 1.

Lemma 4. Lyapunov moments inequality asserts that, for r > s > t > 0,

(EX^{s})^{r−t} ⩽ (EX^{r})^{s−t}(EX^{t})^{r−s}.

ON REGULARLY VARYING MOMENTS FOR POWER SERIES DISTRIBUTIONS 255

1. Results

1.1. The above Theorem 2 has many applications in real or complex analysis (cf. [4]). We shall apply it here to derive the following theorem on regularly varying moments for discrete laws.

Theorem 3. Let a discrete law G be given by P (X_{n} = k) = p_{nk} ⩾ 0, ∑_{k} p_{nk} = 1. If EX_{n} → ∞ and EX_{n}^{β} ∼ C_{β}(EX_{n})^{β} (n → ∞) for β ∈ (0, B], B > 1, C_{β} > 0 then, for an arbitrary slowly varying function ℓ(·), the asymptotic relation

EX_{n}^{β}ℓ(X_{n}) ∼ C_{β}(EX_{n})^{β}ℓ(EX_{n}) (n → ∞),

holds

a. for each β ∈ (0, B − 1];

b. for each β ∈ (B − 1, B], if EX_{n}^{B+1} exists and EX_{n}^{B+1} = O((EX_{n})^{B+1}) (n → ∞).

Proof. Putting A_{nk} := p_{nk}k^{β}/C_{β}(EX_{n})^{β}, we find out that conditions (i) and (ii) of Theorem 2 are satisfied. For β ∈ (0, B − 1], we obtain

t_{n}(1) = EX_{n}^{β+1}/C_{β}(EX_{n})^{β} ∼ (C_{β+1}/C_{β})EX_{n} → ∞ (n → ∞).

Also,

t_{n}(−β/2) = EX_{n}^{β/2}/C_{β}(EX_{n})^{β} ∼ (C_{β/2}/C_{β})(EX_{n})^{−β/2} = O(t_{n}(1))^{−β/2} (n → ∞).

Therefore, the conditions of Theorem 2 are satisfied with A = 1, a = β/2 and the result for β ∈ (0, B − 1] follows. □

For the case β ∈ (B − 1, B], we need the following

Lemma 5. Under the condition b of Theorem 3, we have

(4)  EX_{n}^{β+1} = O((EX_{n})^{β+1}) (n → ∞),

for each β ∈ (B − 1, B].

Proof. Indeed, applying Lyapunov moments inequality (Lemma 4) with r = B + 1, s = β + 1, t = B, we get

\begin{aligned}EX_{n}^{β+1} & ⩽ (EX_{n}^{B})^{B−β}(EX_{n}^{B+1})^{β+1−B} \\ & = O((EX_{n})^{B(B−β)}(EX_{n})^{(B+1)(β+1−B)}) = O((EX_{n})^{β+1}). \end{aligned}

Now, by Jensen’s inequality EX_{n}^{β+1} ⩾ (EX_{n})^{β+1} i.e., t_{n}(1) ⩾ EX_{n}/C_{β} → ∞ (n → ∞).

Also, by (4),

t_{n}(−β/2) ∼ (C_{β/2}/C_{β})(EX_{n})^{−β/2} = O((EX_{n}^{β+1}/(EX_{n})^{β})^{−β/2}) = O((t_{n}(1))^{−β/2}).

Therefore, the conditions of Theorem 2 are satisfied and we get

EX_{n}^{β}ℓ(X_{n}) ∼ C_{β}(EX_{n})^{β}ℓ(EX_{n}^{β+1}/C_{β}(EX_{n})^{β}) (n → ∞),

for each β ∈ (B − 1, B].

But, since

EX_{n}/C_{β} ⩽ \frac{EX_{n}^{β+1}}{C_{β}(EX_{n})^{β}} = O(EX_{n}) (n → ∞),

it follows by the uniform convergence theorem for slowly varying functions (cf. [1, p.6]), that

ℓ(\frac{EX_{n}^{β+1}}{C_{β}(EX_{n})^{β}} ) ∼ ℓ(EX_{n}) (n → ∞).  □

1.2. We turn back now to the asymptotic evaluation of regularly varying moments for power series distributions. Using Theorem 3 above, it will be shown that this evaluation is equivalent to the following theorem on moments of power series distributions.

Theorem 4. For each α > 0, we have E_{w}X^{α} ∼ (E_{w}X)^{α} (w → ∞).

For the generating entire function f(w) = ∑ a_{k}w^{k} ∈ A_{ρ}, recall (1) and (2):

E_{w}X = ∑ ka_{k}w^{k}/f(w) = wf′(w)/f(w);
E_{w}X^{m} = ∑ k^{m}a_{k}w^{k}/f(w) = f_{m}(w)/f(w).

The proof of Theorem 4 requires some preliminary lemmas.

Lemma 6. The expectation E_{w}X is a monotone increasing and unbounded function in w.

Proof. Since

w \frac{d}{dw} (E_{w}X) = E_{w}X^{2} − (E_{w}X)^{2} > 0,

we conclude that E_{w}X is a monotone increasing function in w. If it is bounded, then there exists a d > 0 such that E_{w}X < d for each w > 0. By (1) we get f′(w)/f(w) < d/w, and integrating we find f(w) = O(w^{d}). Hence in this case f is a polynomial, which contradicts our assumption that f is a transcendental entire function. □

Lemma 7. For m ∈ N, f_{m}(w) ∼ w^{m}f^{(m)}(w) (w → ∞).

Proof. Note that f ∈ A_{ρ} implies f^{(m)}, f_{m} ∈ A_{ρ}, m = 1, 2, · · ·. Since, for fixed m ∈ N,

f_{m}(w) = ∑ k^{m}a_{k}w^{k}; w^{m}f^{(m)}(w) = ∑_{k⩾m} k(k − 1) ⋯ (k − m + 1)a_{k}w^{k};

k(k − 1) ⋯ (k − m + 1) ∼ k^{m} (k → ∞),

the result follows by Lemma 2. □

Lemma 8. For each m ∈ N we have E_{w}X^{m+1}/E_{w}X^{m} ∼ E_{w}X (w → ∞).

ON REGULARLY VARYING MOMENTS FOR POWER SERIES DISTRIBUTIONS 257

Proof. Applying Theorem 1, we obtain

\frac{E_{w}X^{2}}{(E_{w}X)^{2}} → 1 (w → ∞),  (5)

because

\frac{E_{w}X^{2}}{(E_{w}X)^{2}} − \frac{1}{E_{w}X} = \frac{f(w)f′′(w)}{(f′(w))^{2}} → 1 (w → ∞),

and, by Lemma 6, 1/E_{w}X → 0.

Since Theorem 1 is valid for each f ∈ A_{ρ} and f^{(m)} ∈ A_{ρ}, m = 1, 2, ⋯, replacing f by f^{(m)}, we get

(6) \frac{f^{(m+1)}(w)f^{(m−1)}(w)}{(f^{(m)}(w))^{2}} → 1 i.e. \frac{f^{(m+1)}(w)}{f^{(m)}(w)} ∼ \frac{f^{(m)}(w)}{f^{(m−1)}(w)} (w → ∞).

Hence by Lemma 7 and (6),

\begin{aligned}\frac{E_{w}X^{m+1}}{E_{w}X^{m}} = \frac{f_{m+1}(w)}{f_{m}(w)} & ∼ \frac{w^{m+1}f^{(m+1)}(w)}{w^{m}f^{(m)}(w)} ∼ \frac{w^{m}f^{(m)}(w)}{w^{m−1}f^{(m−1)}(w)} \\ & ∼ \frac{f_{m}(w)}{f_{m−1}(w)} = \frac{E_{w}X^{m}}{E_{w}X^{m−1}} , n ∈ N. \end{aligned}

Therefore,

\frac{E_{w}X^{m+1}(w)}{E_{w}X^{m}} ∼ \frac{E_{w}X^{m}}{E_{w}X^{m−1}} ∼ ⋯ ∼ \frac{E_{w}X^{2}}{E_{w}X} ∼ E_{w}X (w → ∞).  □

A simple consequence of the previous lemma is the following:

Lemma 9. For each m ∈ N, we have E_{w}X^{m} ∼ (E_{w}X)^{m} (w → ∞).

Proof. Indeed,

E_{w}X^{m} = (E_{w}X) ∏_{k=1}^{m−1}(E_{w}X^{k+1}/E_{w}X^{k}) ∼ (E_{w}X)^{m} (w → ∞).

For the rest of the proof of Theorem 4 we apply Lemma 4.

Let m > α > m − 1, m ∈ N. Then Lyapunov’s inequality and Lemma 9 give

\begin{aligned}E_{w}X^{α} ⩽ (E_{w}X^{m})^{α−m+1}(E_{w}X^{m−1})^{n−α} & ∼ (E_{w}X)^{m(α−m+1)}(E_{w}X)^{(m−1)(m−α)} \\ & = (E_{w}X)^{α}. \end{aligned}

Hence

lim sup_{w→∞} \frac{E_{w}X^{α}}{(E_{w}X)^{α}} ⩽ 1.

Now, let r = m+1, s = m, t = α. We get (E_{w}X^{m})^{m+1−α} ⩽ (E_{w}X^{α})(E_{w}X^{m+1})^{n−α}, i.e.,

\begin{aligned}E_{w}X^{α} ⩾ (E_{w}X^{m})^{m+1−α}(E_{w}X^{m+1})^{α−m} & ∼ (E_{w}X)^{m(m+1−α)}(E_{w}X)^{(m+1)(α−m)} \\ & = (E_{w}X)^{α}. \end{aligned}

Therefore,

lim inf_{w→∞} \frac{E_{w}X^{α}}{(E_{w}X)^{α}} ⩾ 1,

and this concludes the proof of Theorem 4. □

1.3. Combining the last two theorems, we finally obtain a theorem on regularly varying moments for power series distributions.

Theorem 5. For a power series distribution generated by an entire function f(w) = ∑ a_{k}w^{k} ∈ A_{ρ}, we have

E_{w}X^{α}ℓ(X) ∼ (E_{w}X)^{α}ℓ(E_{w}x), α > 0 (w → ∞),

i.e.,

∑ k^{α}ℓ(k)a_{k}w^{k}/f(w) ∼ (wf′(w)/f(w))^{α}ℓ(wf′(w)/f(w)) (w → ∞),

where ℓ(·) is an arbitrary slowly varying function .

As an example we take the well-known Poisson distribution. Applying Theorem 5, we obtain

Theorem 6. For the Poisson law defined by P (X = k) = \frac{λ^{k}}{k!} e^{−λ}, λ > 0, k = 0, 1, 2, ⋯, we have

EX^{α}ℓ(X) := ∑ k^{α}ℓ(k)\frac{λ^{k}}{k!} e^{−λ} ∼ λ^{α}ℓ(λ), α > 0 (λ → ∞).

References

[1] N. H. Bingham, C. M. Goldie, J. I. Teugels, Regular Variation, Cambridge Univ. Press, 1987.

[2] N. L. Johnson, S. Kotz, A. W. Kemp, Univariate Discrete Distributions, John Wiley and Sons, 1993.

[3] G. Pólya, G. Szégö, Aufgaben und Lehrsätze aus der Analysis, Springer-Verlag, 1964.

[4] S. Simić, On complex-valued matrix transforms with slowly varying sequences, Ind. J. Pure Appl. Math., to appear.

[5] S. Simić, A theorem concerning entire functions with positive Taylor coefficients, SIAM Online Problems, www.siam.org, 2005.

Matematički institut SANU (Received 12 05 2006) Knez Mihailova 35 (Revised 19 10 2006) 11000 Beograd Serbia

    ssimic@mi.sanu.ac.yu