General Nörlund transforms and power series methods

by

Rüdiger Kiesel

1 Introduction and main result

Let us start by recalling the following summability methods: We say a sequence (s_{n}) of complex numbers is summable to s by the

(i) Cesàro-method s_{n} → s(C_{α}) for α > 0 if

\frac{1}{\binom{n+α}{n}} ∑_{k=0}^{n} \binom{n − k + α − 1}{n − k}s_{k} → s (n → ∞).

(ii) Euler-method s_{n} → s(E_{α}) for α > 0 if

\frac{1}{(α + 1)^{n}} ∑_{k=0}^{n} \binom{n}{k}α^{n−k}s_{k} → s (n → ∞).

(iii) Abel-method s_{n} → s(A) if

(1 − t) ∑_{n=0}^{∞} s_{n}t^{n} → s (t → 1−).

(iv) Borel-method s_{n} → s(B) if

e^{−t} ∑_{n=0}^{∞} \frac{s_{n}}{n!} t^{n} → s (t → ∞).

The Cesàro- and Abel-method resp. the Euler- and Borel-method are known to be closely related, see [13,23,25]. Recall especially the following two classical results, see [13, Theorem 92; 15] resp. [1, Theorem 0; 16]

THEOREM A

(i) For 0 < α ≤ β:

s_{n} → s(C_{α}) implies s_{n} → s(C_{β}) implies s_{n} → s(A). (Briefly: (C_{α}) ⊆ (C_{β}) ⊆ (A) ).

(ii) For s_{n} = O (1) s_{n} → s(A) implies s_{n} → s(C_{α}) for all α > 0 .

THEOREM B

(i) For 0 < α ≤ β: (E_{α}) ⊆ (E_{β}) ⊆ (B) .

(ii) For s_{n} = O (1) s_{n} → s(B) implies s_{n} → s(E_{α}) for all α > 0 .

Statement (i) of both theorems is an implication of Abelian character. It say, that the applicability of the Cesàro- resp. Euler-methods grows wider with increasing α, in the sense that the sets of summable sequences are increasing. For α → ∞ these sets tend in a certain sense to the set of (A) resp. (B) summable sequences.

Statement (ii) is of Tauberian character and really says, that for bounded sequences the class of Cesàro-methods and Abels method resp. the class of Euler-methods and Borels method are equivalent, briefly (C_{α}) ≈ (A) resp. (E_{α}) ≈ (B) .

Discussion of the properties of power series methods, especially Tauberian theorems, i.e. implications from convergence of the power series method to convergence of some weaker summability methods or ordinary convergence, has a long history, see e.g. the books [13,23,25], and has recently found new attention in the papers [11,14,24] and [8,18,19,20].

In the latter papers general power series methods are discussed. These methods are defined in the following way. Assume that (p_{n}) is a sequence of reals with

(1.1)  \{ \begin{matrix}p_{0} > 0, p_{n} ≥ 0, n ∈ N I , such that the power series \\ p(t) = ∑_{n=0}^{∞} p_{n}t^{n} has radius of convergence R > 0. \end{matrix}

For complex sequences (s_{n}) with increments a_{n} = s_{n} − s_{n−1}, n ≥ 0, s_{−1} = 0 , we consider the power series method of summability (P) , where we say

(1.2)  \{ \begin{matrix}s_{n} → s(P), & if p_{s}(t) = ∑_{n=0}^{∞} s_{n}p_{n}t^{n} converges for |t| < R \\ & and if σ_{p}(t) = \frac{p_{s}(t)}{p(t)} → s as t → R − . \end{matrix}

The power series methods contain the (J_{p}) - or Abel-type methods with R = 1 and Abel’s method, p_{n} = 1 , as a special case. The (B_{p}) - or Borel-type methods with R = ∞ and Borel’s method, p_{n} = 1/n! , as a special case are similarly covered.

Since we can use p_{n}R^{n} as weights in case 0 < R < ∞, we only have to deal with the two cases R = 1 and R = ∞. Furthermore we require the (P) -methods to be regular, i.e. s_{n} → s (n → ∞) implies s_{n} → s(P) . By Theorem 5 in [13], we have regularity if and only if

(1.3) \{ \begin{matrix}(A) & P_{n} = ∑_{k=0}^{n} p_{k} → ∞, (n → ∞), in case R = 1, \\ (B) & p(t) is not a polynomial, i.e. \\ & p_{n} ≠ 0 for infinitely many n in case R = ∞. \end{matrix}

During the discussion of general power series methods the sequence of maximal weights” (Δ_{n}) defined by

(1.4) Δ_{n} = inf_{0<t<R} p(t)t^{−n}.

played a decisive role, especially in formulating Tauberian conditions. For a general discussion the weights p_{n} cannot be too irregular, so we have to impose a regularity condition.

(1.5) p_{n} ∼ q_{n} = exp{−g(n)}, (n → ∞),

with a real function g satisfying the conditions

(C) \{ \begin{matrix}g ∈ C_{2}[x_{0}, ∞) for some x_{0} ∈ N I , \\ g′′(x) is positive and decreasing, with lim_{x→∞} g′′(x) = 0, and \\ G(x) := x^{2}g′′(x) is increasing on [x_{0}, ∞) \end{matrix}

Under (1.5) all ‘smooth’ sequences (p_{n}) are included with p_{n} growing faster than some positive power of n. So the Abel-method is not directly included, but we treat it as belonging to the limiting case q_{n} = L(n) , with L(.) an increasing, slowly varying function, i.e L ∈ SV , see [4], §1.2 for the definition.

Under these conditions we have the relationship

(1.6) Δ_{n} = ϕ(n) p_{n} (n → ∞),

with ϕ(t) a suitable, continuous function.

The aim of this note is to find matrix methods, which have similar properties related to general power series methods as our paradigms Cesàro- and Euler-methods have. We especially want to establish results similar to Theorems A and B for general power series methods. Results as in part (ii) are very useful in proving Tauberian theorems, since general Tauberian conditions always imply the boundedness of the sequence (s_{n}) in question, see [18, Theorem 1], and then one can work with the matrix method, which is easier to handle.

To this end we introduce the (N, p^{∗α}, p) -method of summability for α ∈ N I . These methods are defined in the following way:

Let (p_{n}) be a sequence of reals satisfying (1.1) and

(1.7) p_{n}^{∗α} > 0, n = 0, 1, 2, … α ∈ N I − {1},

where p_{n}^{∗α} is defined by

(1.8) p_{n}^{∗1} := p_{n}, n = 0, 1, 2, … and p_{n}^{∗(α+1)} := ∑_{k=0}^{n} p_{n−k}^{∗α}p_{k}

Then we say s_{n} → s (N, p^{∗α}, p) if

(1.9) σ_{n}^{α} := \frac{1}{p_{n}^{∗(α+1)}} ∑_{k=0}^{n} p_{n−k}^{∗α}p_{k}s_{k} → s (n → ∞).

Remark 1: For α > 0 a noninteger, we define p_{n}^{∗α} as the n-th coefficient in the power series expansion of p(t)^{α} at 0 , which is defined since p_{0} ≠ 0 . We use the above definition, since we concentrate on the integer case only.

We can now state our main theorem

THEOREM 1

Let (p_{n}) be a sequence of reals satisfying (1.1) and (1.5), then

(i) for 1 ≤ α ≤ β, α, β ∈ N I , we have

(N, p^{∗α}, p) ⊆ (N, p^{∗β}, p) ⊆ (P).

(ii) s_{n} = O (1) and s_{n} → s (P) imply s_{n} → s (N, p^{∗α}, p) for all α ∈ N I .

Remark 2

(i) Since p_{n} ≡ 1 (as the limiting case) resp. p_{n} = 1/n! ∼ exp\{(n + \frac{1}{2}) log n − n + \frac{1}{2} log(2π)\} by Stirling’s formula in the Abel resp. Borel case Theorems A and B are included.

(ii) Under the condition s_{n} = O (1) the power series methods (P) , with weight sequence (p_{n}) , and (Q) , with weight sequence (q_{n}) , are equivalent, see [19] Lemma 2, and the same is true for the generalized Nörlund means (N, p, p) and (N, q, q) .

(iii) The inclusions in Theorem 1 can be obtained under weaker conditions, see §2, Proposition 1.

(iv) The following methods are also covered, see [17,18,19,20] for further discussion and references Abel–type

(i) (J_{p}) Methods with regularly varying weights (A_{R_{α}}) : p_{n} = n^{α}L(n), α > 0, L(.) ∈ SV , Δ_{n} ∼ cn with α = 0, L(t) ≡ 1 as the limiting case being the classical Abel method (A) .

(ii) Methods with rapidly increasing weights (A_{p}) : for example p_{n} = exp\{n^{α}\}, 0 < α < 1, Δ_{n} ∼ cn^{1− \frac{α}{2}} p_{n}.

Borel–type

(iii) Generalized Borel-methods (B(α, β)) :

p_{n} = \frac{α}{Γ(αn + β)}, t replaced by t^{α}, α, β > 0, Δ_{n} ∼ c\sqrt{n}p_{n} with

α = β = 1 the classical Borel-method (B) .

(iv) (B_{p}) –Methods:

for example p_{n} = exp\{−n^{β}\}, 1 < β < 2, Δ_{n} ∼ cn^{1− \frac{β}{2}} p_{n}.

We start out in §2 to discuss the relation of the (N, p^{∗α}, p) - methods to other summability methods, especially the weighted mean methods and methods of moving averages. We also prove a slight generalization of Theorem 1(i). Furthermore we give an account of the historical context of the generalized Nörlund methods. In §3 we concentrate on the case ϕ(t) = o (t) and generalize a classical theorem on the connection between Cesàro- and Euler-methods. In §4 we complete the proof of Theorem 1 and finally in §5 we give applications of the use of (N, p^{∗α}, p) - methods in probablitity theory and introduce some new families of summability methods comparable to the class of circle-methods, see [21].

2 Definitions and preliminary results

The first definition of the generalized Nörlund transform (N, a, b) was given by D. Borwein in [6] and is as follows: Let (a_{n}) and (b_{n}) be sequences of real numbers such that

(a ∗ b)_{n} = ∑_{k=0}^{n} a_{n−k}b_{k} ≠ 0, n = 0, 1, …

Given a sequence (s_{n}) of complex numbers, its sequence (σ_{n}) of (N, a, b) -transforms is defined by

(2.1) σ_{n} = \frac{1}{(a ∗ b)_{n}} ∑_{k=0}^{n} a_{n−k}b_{k}s_{k}

and we say s_{n} → s(N, a, b) , if σ_{n} → s (n → ∞) .

Using the Toeplitz theorem [13, Theorem 2] Borwein gave the necessary and sufficient conditions for the regularity of the (N, a, b) -methods,

(2.2)

(i) ∑_{k=0}^{n} |a_{n−k}b_{k}| = O((a ∗ b)_{n})
(ii) \frac{a_{n−k}b_{k}}{(a ∗ b)_{n}} → 0 (n → ∞) for each integer k.

A systematic discussion of the relation of the (N, a, b) -methods to weighted means (N, 1, b) and Nörlund means (N, a, 1) , where 1 stands for the sequence (1, 1, …) , was given by Das in [9,10]. Although Borwein and Das observed that Cesàro- and Euler-methods are special cases of (N, a, b) -methods, none of them mentioned the possibility of generating them as (N, p^{∗α}, p) -methods.

We now start our investigation with a slight generalization of Theorem 1(i).

PROPOSITION 1

Let α ≤ β, α, β ∈ N I and (p_{n}) be a sequence of weights satisfying (1.1) and (1.7).

(i) If (p_{n}) generates a regular (P) - method, then

(N, p^{∗α}, p) ⊆ (P).

(ii) If (p_{n}) generates regular (N, p^{∗α}, p) -methods for all α ∈ N I , then

(N, p^{∗α}, p) ⊆ (N, p^{∗β}, p).

Proof: To show the inclusion (i), we first show that each sequence (p_{n}^{∗α}) generates a regular (P) -method. In case R = 1, p(t)^{α} still has convergence radius R ≥ 1 and since (p∗p)_{n} ≥ p_{0}p_{n}, we have

∑_{k=0}^{n}(p ∗ p)_{k} ≥ p_{0} ∑_{k=0}^{n} p_{k} = p_{0}P_{n} → ∞ (n → ∞).

Therefore R = 1 and the method is regular.

In case R = ∞ condition (1.7) implies the regularity.

Now let s_{n} → s (N, p^{∗α}, p) . By the regularity of all methods follows that

σ_{p}(t) = \frac{p_{s}(t)}{p(t)} = \frac{∑_{n=0}^{∞} p_{n}^{(α+1)} (\frac{1}{p_{n}^{(α+1)}} ∑_{k=0}^{n} p_{n−k}^{∗α}p_{k}s_{k}) t^{n}}{∑_{n=0}^{∞} p_{n}^{(α+1)}t^{n}} → s, (t → R−).

For the second inclusion we use the Toeplitz theorem to show that

A = (N, p^{∗β}, p)(N, p^{∗α}, p)^{−1}

is regular. The inverse method (N, p^{∗α}, p)^{−1} can be obtained by using the power

series p(t) = ∑_{n=0}^{∞} p_{n}t^{n}. Since p_{0} ≠ 0, p(t)^{−α} has a power series expansion in

some neighbourhood of 0 , say p(t)^{−α} = c(t) = ∑_{n=0}^{∞} c_{n}t^{n}. Then

(N, p^{∗α}, p)_{(n,k)}^{−1} = \{ \frac{c_{n−k}p_{k}^{∗(α+1)}}{p_{n}} ; 0 for \begin{matrix}k ≤ n \\ n < k. \end{matrix}

For k ≤ n we hence have

A_{(n,k)} = a_{n,k} = \frac{1}{p_{n}^{∗(β+1)}} ∑_{ν=k}^{n} p_{n−ν}^{∗β}p_{ν} \frac{1}{p_{ν}} c_{ν−k}p_{k}^{∗(α+1)}

= \frac{p_{k}^{∗(α+1)}}{p_{n}^{∗(β+1)}} ∑_{ν=k}^{n} p_{n−ν}^{∗β}c_{ν−k}

= \frac{p_{k}^{∗(α+1)}p_{n−k}^{∗(β−α)}}{p_{n}^{∗(β+1)}} ,

where we used the coefficient formula for power series. Since the (N, p^{∗α}, p) - matrices are lower diagonal matrices so are the inverse matrices and therefore A, hence a_{n,k} = 0 , if k > n. Now a_{n,k} ≥ 0 for all n, k ≥ 0 and

∑_{k=0}^{n} a_{n,k} = \frac{1}{p_{n}^{∗(β+1)}} ∑_{k=0}^{n} p_{k}^{∗(α+1)}p_{n−k}^{∗(β−α)} = \frac{p_{n}^{∗(β+1)}}{p_{n}^{∗(β+1)}} = 1

by the coefficient formula. Therefore the only Toeplitz condition, which remains to be verified is that for each k ∈ N I

\frac{p_{n−k}^{∗(β−α)}}{p_{n}^{∗(β+1)}} → 0, (n → ∞).

By the regularity of the (N, p^{∗γ}, p) -methods, γ ∈ N I , follows

\frac{p_{n−k}^{∗(β−α)}}{p_{n}^{∗(β+1)}} = \frac{p_{n−k}^{∗(β−α)}}{p_{n−k}^{∗(β−α+1)}} × \frac{p_{n−k}^{∗(β−α+1)}}{p_{n−k}^{∗(β−α+2)}} × … × \frac{p_{n−k}^{∗β}}{p_{n}^{∗(β+1)}} → 0, (n → ∞)

Remark 3:

(i) Parts of the proof can also be deduced using results of Das in [9, Theorem 1] and [10, Theorem 1].

(ii) Das also discussed in [10] the the Relations of (N, a, b) methods to each other. He particulary showed that if

\begin{matrix}p_{n+1} ≥ p_{n} or \\ p_{n} ≥ p_{n+1} and P_{n} = O((p ∗ p)_{n}) \end{matrix} \} then (N, 1, p) ⊆ (N, p, p).

Under the condition p_{n+1}/p_{n} ≤ p_{n+2}/p_{n+1} he also showed the reserve inclusion. In case R = 1 we can relax this condition.

LEMMA

Let (p_{n}) be a non-increasing sequence of reals satisfying (1.1) with R = 1 . Then

(N, p, p) ⊆ (N, 1, p)

Proof:

Let p(t)^{−1} = c(t) = ∑_{n=0}^{∞} c_{n}t^{n} and C_{n} = ∑_{k=0}^{n} c_{k}. Then C_{0} = c_{0} > 0 and by

induction C_{n} ≥ 0 , since

0 = ∑_{k=0}^{n} c_{k}p_{n−k} = C_{n}p_{0} + ∑_{k=0}^{n−1} C_{k} \underbrace{(p_{n−k} − p_{n−k−1})}_{≤0} .

Now one can use the Toeplitz theorem to show that

(N, 1, p)(N, p, p)_{n,k}^{−1} = \frac{(p ∗ p)_{k}C_{n−k}}{P_{n}}

is regular.

While the (N, 1, p) and the (N, p, 1) are obviously related to the (N, p, p) - methods their use in proofs of theorems on (N, p, p) means is limited to the case t = O (ϕ(t)) . In case ϕ(t) = o (t) the methods of moving averages are more helpful. These methods are very important in probability theory, see [3,17], and will be needed in the proof of Theorem 1(ii).

We say a sequence (s_{n}) of complex numbers is summable to s by the method of moving averages (R_{ϕ}) , where ϕ(.) is a self-neglecting function, ϕ ∈ SN , see [3,4 §2.11], briefly s_{n} → s(R_{ϕ}) if

(2.3)  \frac{1}{uϕ(n)} ∑_{n≤k<n+uϕ(n)} s_{k} → s, ∀u > 0, (n → ∞).

These methods are related to the Cesàro-method by the following

PROPOSITION 2

The following statements are equivalent

(i) \frac{1}{n + 1} ∑_{k=0}^{n}(s_{k} + ϵ_{k}) = s + o (\frac{ϕ(n)}{n}) (n → ∞) for some ϵ_{n} → 0.

(ii) s_{n} → s(R_{ϕ}).

Proof : For ϕ(t) = t^{γ}, 0 < γ < 1 this is Theorem 3 in [5]. The general case follows from Theorem 2 in [2], since this theorem is applicable also for ϕ ∈ SN. Compare also the proof of Theorem 1 in [3].

We conclude the section by computing the asymptotics of

p_{n}^{∗2} = (p ∗ p)_{n} = ∑_{k=0}^{n} p_{n−k}p_{k}

in two important cases .

(1.5 A) p_{n} ∼ n^{σ}L(n)

with σ > −1, L(.) slowly varying.

This gives an Abel-type method of polynomial growth with convergence radius R = 1 . If we have lim_{x→∞} G(x) = σ > 0 we are in the general case (1.5) with σ > 0 ,

compare [8].

(1.5 B) p_{n} ∼ exp{−g(n)} with lim_{x→∞} G(x) = ∞.

These are the Abel-type methods with rapidly increasing weights and the Borel-type case. In this case the function ϕ(.) of (1.6) is selfneglecting.

A discussion of these conditions with further implications on the behaviour of the weights and the generated power series methods is given in [8,18,19,20]. Especially the asymptotic behaviour of p_{n}t^{n}/p(t) has been studied in detail under these conditions. Since the methods used there are applicable for our problem, we only outline the proofs.

PROPOSITION 3

Let (p_{n}) be a sequence of reals satisfying (1.1) and (1.5) then

(p ∗ p)_{n} ∼ \{ \begin{matrix}n^{2σ+1}L(n)^{2}B(σ + 1, σ + 1) & & (A) \\ & in case \\ \sqrt{\frac{π}{g′′(n/2)}} exp\{−2g(n/2)\} & & (B) \end{matrix}

(B(σ, σ) denotes the Beta-Integral ∫_{0}^{1} t^{σ−1}(1 − t)^{σ−1} dt.) Proof: Case (A): compare [14]

(p ∗ p)_{n} ∼ ∫_{0}^{n}(n − x)^{σ}x^{σ}L(n − x)L(x) dx

= n^{2σ+1}L(n)^{2} ∫_{0}^{1}(1 − x)^{σ}x^{σ} \frac{L(n(1 − x))L(x)}{L(n)^{2}} dx

∼ n^{2σ+1}L(n)^{2}B(σ + 1, σ + 1) (n → ∞).

Case (B): Compare [8,19,20] We use the so-called saddle point method. Define

\begin{aligned}φ(n, k) = φ(k) & = (−g(n − k) + g(n/2)) − g(k) + g(n/2) \\ & = −1/2(k − n/2)^{2}(g′′(ξ_{1}) + g′′(ξ_{2})) \end{aligned}

by Taylor’s formula with k ≤ ξ_{2} ≤ n/2 ≤ ξ_{1} ≤ n − k. We define now a function ϵ(x) , compare [18,19,20], as

ϵ(x) = x(x^{2}g′′(x))^{− \frac{1}{4}} .

Then we have

\frac{ϵ(x)}{x} → 0 and ϵ(x)\sqrt{g′′(x)} → ∞ (x → ∞).

Furthermore the following basic inequality holds true

(2.4)  |\frac{g(t)}{g(x)} − 1| ≤ \frac{ϵ(x)}{x} for |t − x| ≤ ϵ(x).

Then using the properties of ϵ(.) , especially (2.4), and the basic inequality |1 − e^{−x}| ≤ x, for 0 ≤ x, we obtain

\frac{(p ∗ p)_{n}}{exp\{−g(n/2)\}} = 2 ∫\frac{n}{2} −ϵ( \frac{n}{2} )^{\frac{n}{2}} exp\{φ(n, x)\} dx + o(1)

= ∫_{\frac{n}{2} −ϵ( \frac{n}{2} )}^{\frac{n}{2} +ϵ( \frac{n}{2} )} exp\{−g′′(n/2)(x − n/2)^{2}\} dx

+2 ∫\frac{n}{2} −ϵ( \frac{n}{2} )^{\frac{n}{2}} exp\{−g′′(n/2)(x − n/2)^{2}\}

× \{exp \{−\frac{1}{2}g′′(n/2)(x − n/2)^{2} (\frac{g′′(ξ_{1})}{g′′(n/2)} − 1 + \frac{g′′(ξ_{2})}{g′′(n/2)} − 1)\} − 1\} dx + o(1)

= \sqrt{\frac{π}{g′′(n/2)}} + o(1) .

Remark 4:

(i) The asymptotics show, that in the above cases the (N, p, p) - methods are regular, since p_{n−k}/(p ∗ p)_{n} → 0 (n → ∞) ∀k ∈ N I .

(ii) For higher order convolutions p_{n}^{∗α} asymptotics can be obtained with the same methods. These asymptotics show the regularity of the (N, p^{∗α}, p) - methods.

(iii) If we assume p_{n}/p_{n+1} to be nondecreasing we have

p_{n−k}p_{k} ≤ p_{n−(k+1)}p_{k+1} if k ≤ n/2 and p_{n−k}p_{k} ≥ p_{n−(k+1)}p_{k+1} if k ≥ n/2.

Therefore the maximal weight is

\frac{p_{n/2}p_{n/2}}{(p ∗ p)_{n}} ,

which is asymptotically equal to p_{n}/Δ_{n} the maximal weight in the power series method (P) . Observe that q_{n}/q_{n+1} is nondecreasing, where (q_{n}) is the the exact weight sequence in (1.5).

(iv) In case (1.5A) with nonincreasing (p_{n}) , i.e −1 < σ ≤ 0 , we have by Remark 3 and the Lemma, that (N, 1, p) summability is equivalent to (N, p, p) summability, if cP_{n} ≤ (p ∗ p)_{n} ≤ P_{n}, with some constant c > 0 . The asymptotics of Proposition 3 and and Lemmas 3.2 and 3.4 in [14] show that this can only be possible, if we are in the classical Cesàro-case.

3 The relation of Cesàros to the (N, p^{∗α}, p) -methods

We start by recalling a classical result of Knopp, see [13, Theorem 149]

THEOREM CE

If

\frac{1}{n + 1} ∑_{k=0}^{n} s_{k} = s + o (1/\sqrt{n}) , (n → ∞)

then s_{n} → s(E_{α}) for every α > 0 .

For weights (p_{n}) satisfying condition (1.5B) with p_{n}/p_{n+1} nondecreasing, we can generalize this result. Observe that in this cases we furthermore have ϕ(.) ∈ SN (with ϕ(.) as in (1.6)).

THEOREM 2

Let (p_{n}) be a sequence of weights satisfying (1.5B) and (3.1) p_{n}/p_{n+1} is nondecreasing. Then

\frac{1}{n + 1} ∑_{k=0}^{n} s_{k} = s + o (\frac{ϕ(n)}{n}) , (n → ∞)

implies s_{n} → s(N, p^{∗α}, p) for every α ∈ N I .

Proof: By the inclusion theorem we only have to show the implication for the (N, p, p) - method. Because of regularity and linearity we can suppose s = 0 . Then

(3.2)  ∑_{k=0}^{n} s_{k} = o (ϕ(n)) , (n → ∞).

Furthermore from the results in [8,19,20] we have under (1.5B) that ϕ(n) ∼ \sqrt{(2π)/g′′(n)}, (n → ∞) and hence using Proposition 3

(3.3) (p ∗ p)_{n} ∼ \sqrt{1/2}ϕ(n/2) exp\{−2g(n/2)\}, (n → ∞).

By Remark 4 (iii) and (3.1) we have therefore for the maximal weight in the (N, p, p) -transform

(3.4) \frac{p_{n/2}p_{n/2}}{(p ∗ p)_{n}} ∼ \frac{\sqrt{2}}{ϕ(n/2)}, (n → ∞).

Now let ϵ > 0 be given, we can find a N ∈ N I , such that for n ≥ l ≥ m ≥ N

(3.5)  | ∑_{k=m}^{l} s_{k}| ≤ ϵϕ(l) ≤ ϵϕ(n),

using also the monotonicity of ϕ(.) . We split the (N, p, p) -transform in three parts, namely

σ_{n} = \frac{1}{p_{n}^{∗2}} ∑_{k=0}^{n} p_{n−k}p_{k}s_{k} = \frac{1}{p_{n}^{∗2}} (∑_{k=0}^{N} + ∑_{k=N+1}^{n/2−1} + ∑_{k=n/2}^{n} p_{n−k}p_{k}s_{k}) = ∑_{1} + ∑_{2} + ∑_{3} .

By Abel’s Partial Summation we get

|∑_{2}| ≤ | ∑_{k=N+1}^{n/2−1} s_{k}| \frac{p_{n/2}^{2}}{p_{n}^{∗2}} + \frac{1}{p_{n}^{∗2}} ∑_{k=N+1}^{n/2−1} | ∑_{ν=N+1}^{k} s_{ν}| |p_{n−k}p_{k} − p_{n−k−1}p_{k+1}|

We now use (3.3) and (3.5) and the monotonicity of p_{n−k}p_{k} by (3.1) to find

|∑_{2}| ≤ ϵ + ϵ\frac{ϕ(n/2)}{p_{n}^{∗2}}(p_{n/2}^{2} − p_{n−N}p_{N}) ≤ c_{2}ϵ.

With a similar argument

|∑_{3}| ≤ c_{3}ϵ

follows. Since N is fixed and \frac{p_{n−k}p_{k}}{p_{n}^{∗2}} → 0, (n → ∞), ∀k it follows that

|∑_{1}| ≤ c_{1}ϵ

Since c_{1}, c_{2}, c_{3} are constants the proof is complete.

Corollary With (p_{n}) and ϕ(.) as in Theorem 2 s_{n} → (R_{ϕ}) implies s_{n} → s(N, p^{∗α}, p) for all α ∈ N I .

Proof: With Proposition 2 s_{n} → (R_{ϕ}) implies

\frac{1}{n + 1} ∑_{k=0}^{n}(s_{k} + ϵ_{k}) = s + o (\frac{ϕ(n)}{n}) , (n → ∞)

with some sequence ϵ_{n} → 0, (n → ∞) . Hence we obtain by the Theorem (s_{n} + ϵ_{n}) → s(N, p^{∗α}, p) . Since (N, p^{∗α}, p) is a linear and regular summability method s_{n} → s(N, p^{∗α}, p) follows.

Remark 5:

(i) The result of the Corollary can also be obtained by direct computation. One has to use the asymptotic properties of the (N, p, p) transform and the techniques in [17].

(ii) Also by the inclusion theorem we have (N, p^{∗α}, p) ⊆ (P) . Therefore under the conditions of Theorem 2 (P) -summability is implied.

(iii) Under the condition s_{n} = O (1) we can omit the assumption of monotonicity (3.1).

4 Proof of theorem 1

Proof: The proof of part(i) follows from Proposition 1 and Remark 4(ii). To prove part(ii) we only have to show s_{n} → s(N, p, p) . The other implications follow by inclusion. We distinguish two cases:

(i) If G(x) → σ > 0 , we have the case of polynomial weights. We include the limiting case σ = 0, i.e p_{n} ∼ L(n) with L ∈ SV and nondecreasing. Furthermore since s_{n} = O (1) we can switch to the asymptotic weights q_{n} = n^{σ}L(n) , with σ ≥ 0, L ∈ SV and nondecreasing. We hence have s_{n} → s(Q) , where (Q) means the power series method with weight sequence (q_{n}) . In this situation we use a general theorem for Laplace transforms by Stadtmüller/ Trautner, see [24], to get

s_{n} → s(N, 1, q).

Now, since (q_{n}) increases we have s_{n} → s(N, q, q) by Remark 3. We use once again the condition s_{n} = O (1) to get s_{n} → s(N, p, p) .

(ii) (p_{n}) as in (1.5B), hence G(x) → ∞ (x → ∞) and the asymptotic weightsequence is q_{n} = e^{−g(n)} . Now we use Theorem 1 in Moh [22].

Define s(u) = s_{[u]} if u ≥ 0 and s(u) = 0 if u < 0 . As in Kiesel/Stadtmüller [18] we find that under s_{n} = O (1) s_{n} → s(P) implies

(4.1)  lim_{x→∞} ∫_{−∞}^{∞} K (\frac{x − t}{ϕ(x)}) s(t) \frac{dt}{ϕ(x)} = s

with K(x) = 1/\sqrt{2π} exp\{−x^{2}/2\}. The conditions of Theorem 1 of Moh, i.e. K(x) ∈ L^{1}(−∞, ∞) , the Fourier transform of K ≠ 0 for any real argument and ϕ(.) ∈ SN , are satisfied. It follows that if we define

H(x) = \{ \begin{matrix}\frac{1}{ϵ}, \\ 0, \end{matrix} if \begin{matrix}x ∈ (−ϵ, 0), \\ x ∉ (−ϵ, 0) \end{matrix}

that

(4.2) lim_{x→∞} ∫_{−∞}^{∞} H (\frac{x − t}{ϕ(x)}) s(t)\frac{dt}{ϕ(x)} = lim_{x→∞} \frac{1}{ϵϕ(x)} ∑_{x≤k<x+ϵϕ(x)} s_{k} = s.

This means s_{n} → s(R_{ϕ}) . By the Corollary to Theorem 2 and the monotonicity of q_{n}/q_{n+1} s_{n} → s(N, q, q) follows. Using again that s_{n} = O (1) we can conclude that s_{n} → s(N, p, p) .

Remark 6 We consider the case ϕ(t) = o (t) only.

(i) If we use s_{n} − s_{n−1} = O (ϕ(n)/n) as additional condition in the setting of Theorem 1 the convergence of s_{n} to s follows. To see that let ϵ tend to 0 in (4.2).

(ii) Under s_{n} = O (1) we have the following equivalences: (R_{ϕ}) ⇔ (N, p^{∗α}, p) ⇔ (P) , compare Bingham [1] for the Euler-Borel-case.

(iii) It is also possible to obtain a proof of part (ii) by direct computation using the asymptotic properties of the (N, p, p) and (P) -transforms.

5 Applications and complements

(i) The (N, p, p) -methods are the natural methods to be used as general matrix summability methods in the following characterisation of moment conditions by summability methods, see [17] Theorem.

Let X, X_{0}, X_{1}, … be i.i.d. random variables and (P) a power-series method with weight sequence (p_{n}) satisfying (1.5). Then the following statements are equivalent.

(i) E(ϕ^{−1}(|X|)) < ∞, E(X) = μ;

(ii) X_{n} → μ(N, p^{∗α}, p) a.s. ∀α ∈ N I ;

(iii) X_{n} → μ(P)a.s.

In the proof for ϕ(t) ≈ t the (C_{α}) -methods were used and for ϕ(t) = o (t) the Riesz-methods (R, exp{Φ(n)}, 1) with Φ(x) = ∫_{1}^{x} d t/ϕ(t) . But (R, exp{Φ(n)}, 1) ⇔ (R_{ϕ}) by Lemma 3 in [17] and (R_{ϕ}) implies (N, p, p) .

(ii) One can also use the (N, p, p) -methods to obtain new examples for Theorem 1 in Borwein [7], where weakly multipicative sequences of random variables are studied.

(iii) In case ϕ(t) = o (t) and (1.5B) or (1.5C) one can attempt to generalize the well-known Taylor (T) and Meyer-König (S) - methods of summability. Together with Euler-, Borel- and Valiron-method these methods build the family of circle methods, see [21]. One observes that (T_{α}) = (E_{α})^{t} and (S_{α}) is obtained from (T_{α}) by omitting the leading zeros in the (T_{α}) methods. Using this procedure starting with an (N, p, p) -method a (T, p, p) -method and a (S, p, p) -method can be formally obtained. Without being explicitly stated a general Valiron method is already in long use (see the definition in (4.1) and [18]).

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