Power series methods and almost sure convergence

by

Rüdiger Kiesel

1 Introduction and main result

Let ( Ω, Σ, P ) be a probability space and suppose all random variables are defined on this space. Let X, X_{0}, X_{1}, X_{2}, … be independent, identically distributed (i.i.d.) random variables and recall Kolmogorov’s classical law of large numbers and its following extensions for further summability methods by respectively Lai [23] and by Chow [8] and Bingham and Tenenbaum [2]:

THEOREM L

The following are equivalent:

(i) E(|X|) < ∞, E(X) = μ;

(ii) \frac{1}{n + 1} ∑_{k=0}^{n} X_{k} → μ (n → ∞) a.s.; i.e. X_{n} → μ(C_{1}) a.s.;

(iii) (1 − t) ∑_{n=0}^{∞} X_{n}t^{n} → μ (t → 1−) a.s.; i.e. X_{n} → μ(A) a.s.;

where (C_{1}) and (A) denote the classical Cesàro- and Abel-summability methods.

THEOREM C–B–T

The following are equivalent:

(i) E(X) = μ, E(X^{2}) < ∞;

(ii) ∑_{k=0}^{n} \binom{n}{k}p^{k}(1 − p)^{n−k}X_{k} → μ (n → ∞) a.s.;

i.e., X_{n} → μ(E_{p}) a.s. for all (some) p ∈ (0, 1) ;

(iii) e^{−t} ∑_{n=0}^{∞} \frac{X_{n}}{n!} t^{n} → μ (t → ∞) a.s.; i.e. X_{n} → μ(B) a.s.

(iv) \frac{1}{u\sqrt{n}} ∑_{k=n}^{n+u\sqrt{n}} X_{k} → μ ∀ u > 0, (n → ∞) a.s.; i.e. X_{n} → μ(R\sqrt{n}) a.s.,

with (E_{p}), (B), (R\sqrt{n}) the classical Euler- ,Borel- and moving–average means, see [13] or §3 below.

Our aim is to extend and unify these theorems to the whole class of power series methods, where (A) and (B) are the best understood examples. For this let us first define a power series method.

We 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}

A widely discussed field is the question of the reverse implication, namely the question, s_{n} → s(P) implies s_{n} → s (n → ∞) . This implication is true only under additional assumptions (the so-called Tauberian conditions) and is called a Tauberian result. Tauberian results have a long history concerning Abel’s and Borel’s method, see [13,24,27] and have recently found new attention for special classes, see [10,15,16]. In a series of papers Kratz,Stadtmüller and other authors [7,19,20,21,22] developed a general theory of Tauberian results for regular power series methods, especially linking Tauberian results for Abel- and Boreltype methods. To formulate our main result, we need some notation and results of Kratz,Stadtmüller (for details see §2 below).

First we introduce the sequence of maximal weights” (Δ_{n}) defined by

(1.4) Δ_{n} = inf_{0<t<R} p(t)t^{−n}.

This sequence is related to the limitation order of the power series method, see [7], and plays a decisive role in formulating Tauberian conditions. In order to prevent the weights p_{n} from being too irregular, we use the following condition

(1.5) Δ_{n} ∼ l(n) p_{n} (n → ∞),

with l(t) a continuous, strictly increasing function. In case l(t) = o (t) , we assume furthermore l(.) as self-neglecting, see [6] §2.11 for the definition, and

(1.5’) p_{n} ∼ exp{g(n)},

with g(.) a logarithmico- exponential function, satisfying certain growtth-conditions, see [10,21,22] and the Remark below.

We are now in the position to state our main result

THEOREM

Let (P) be a regular power series method satisfying (1.5). Then the following statements are equivalent:

(i) E(l^{−1}(|X|)) < ∞;

(ii) (X_{n} − C_{n}) → 0 (S) a.s., where (S) is some weaker summability method depending on (P) ;

(iii) (X_{n} − C_{n}) → 0 (P) a.s.; where C_{n} = E (XI_{[|X|≤l(n)]}) resp., if E(|X|) < ∞, C_{n} = E(X) .

Remark 1

(i) Since Δ_{n} ∼ en resp. Δ_{n} ∼ (\frac{e}{n})^{n} in the Abel resp. Borel case Theorem C

and L–B–T are included.

(ii) Observe that (1.5) together with the regularity condition implies p_{n} = o (Δ_{n}) .

(iii) In case α < 1 we cover the (J_{p}) –case with rapidly increasing weights, i.e. p_{n} ∼ exp{g(n)}, with tg′(t) → ∞ (t → ∞), g′′(t) < 0 and the (B_{p}) –case, i.e. p_{n} ∼ exp{−g(n)}, with g′(t) → ∞ and g′′(t) → ∞ (t → ∞) , see the examples in §2.

(iv) We show in the sequel that our moment-condition covers the whole range of moment-conditions from slightly weaker than first moment to moments of arbitrarly high order.

We start in §2 to give the main results concerning power series methods developed by Kratz,Stadtmüller. In §3 we discuss the connection between power series methods and some classes of weaker summability methods, e.g. the methods of weighted means, the moving average methods and the Riesz-methods. We prove especially a new result of abelian type from moving averages to power series methods, which applies to a large class of such methods, including and generalizing results in [26]. §4 is devoted to the probabilistic parts of our arguments. We generalize a strong law for moving averages, which was first given by Chow in [8]. Finally in §5, we prove our Theorem.

2 Powers series methods and dominated variation

We give some informations on (Δ_{n}) useful in our context. For a detailed discussion and proofs see [7,19,20,22].

LEMMA 1

We have

(2.1) Δ_{n} = p(t_{n})t_{n}^{−n} for n = 0, 1, 2, …, such that

(i) (t_{n}) is non-decreasing and lim_{n→∞} t_{n} = R;

(ii) t_{m}^{n−m} ≤ Δ_{m}/Δ_{n} ≤ t_{n}^{n−m} for n, m ∈ N I ;

(iii) p_{n}/Δ_{n} ≤ 1 and ∑_{n=0}^{∞} \frac{p_{n}}{Δ_{n}} = ∞.

Remark 2

(i) Observe that (iii) is a generalization of the Abel-Dini-theorem.

(ii) In case p_{n} ∼ exp(g(n)) , with some logaritmico-exponential function g(.) , we have under the regularity conditions on the order of growth of g(.) , see Remark 1:

l(n) ∼ c \{\frac{|g′′(n)|}{2π}^{− \frac{1}{2}} \} ,

with c some constant.

LEMMA 2

Assume (1.1). Then the following assertions are equivalent:

(i) P (x) = ∑_{0≤k≤x} p_{k} is dominatedly varying,

i.e lim sup_{λ→∞} \frac{P (λx)}{P (λ)} < ∞ for all (some) x > 1 ;

(ii) Δ_{n} = O (P_{n}) ;

(iii) \frac{np_{n}}{Δ_{n}} = O (1) .

Proof: The following equivalences were obtained in [20, Corollary 3].

(i) ⇔ (ii) ⇔ \frac{p(1 − c/n)}{Δ_{n}} = O (1) for all (some) c > 0, ⇔ \frac{np_{n}}{P_{n}} = O (1) .

That (iii) implies (i) follows by:

\begin{aligned}p(1 − c/n) & ≤ Δ_{n} + \frac{c}{2n} ∑_{k=n}^{∞} Δ_{k}(1 − c/n)^{k} \\ & ≤ Δ_{n} + \frac{cp(1 − c/n)}{2n} ∑_{k=n}^{∞}(1 − c/n)^{k} ≤ Δ_{n} + \frac{1}{2}p(1 − c/n). \end{aligned}

The implication from (i) to (iii) is a trivial application of the basic inequality P_{n} ≤ Δ_{n}.

Examples: Throughout we will illustrate our results on some special methods, which we introduce here. For properties and further references on the methods see [13,24,26].

Abel–type

(i) Generalized Logarithmic Methods (L_{α}) :

p_{n} = \frac{α(log^{α−1}(n + 1))}{n + 1}, α > 0; Δ_{n} ∼ c log n with α = 1 being the classical

logarithmic method (L) .

(ii) Generalized Abel Methods (A_{α}) :

p_{n} = \binom{n + α}{n}; Δ_{n} ∼ cn with α = 0 being the classical Abel method (A) .

(iii) (J_{p}) Methods with regularly varying weights (A_{R_{α}}) : p_{n} = n^{α}L(n), α > −1, L(.) slowly varying, see [25], Δ_{n} ∼ cn with α = 0, L(t) ≡ 1 being the classical Abel method (A) .

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

Borel–type

(v) 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) . (vi) (B_{p}) –Methods: p_{n} = exp\{−n^{β}\}, 1 < β < 2, Δ_{n} ∼ cn^{1− \frac{β}{2}} p_{n}.

3 Definition and properties of auxiliary summability methods

In addition to the power series methods, we shall principally be concerned with three classical important classes of summability methods.

For a complex sequence (s_{n}), s_{n} = ∑_{k=0}^{n} a_{k} recall the weighted mean average (M_{p})

based on a sequence (p_{n}) of nonnegative reals, which satisfies p_{0} > 0 and (1.3A). Then we say s_{n} → s(M_{p}) if

(3.1) \frac{1}{P_{n}} ∑_{k=0}^{n} s_{n}p_{n} → s, (n → ∞).

Secondly recall the Riesz (typical) means (R, λ_{n}, 1) , of order 1 based on a real sequence 0 ≤ λ_{0} < λ_{1} < . . .. We say s_{n} → s(R, λ_{n}, 1) if

(3.2) \frac{1}{t} ∫_{0}^{t} \{ ∑_{λ_{k}≤x} a_{k}\} dx → s, (t → ∞).

Finally, we need the methods of moving averages (R_{ϕ}) , where ϕ(.) is a selfneglecting function. We say s_{n} → s(R_{ϕ}) if

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

The relations of this methods to each other are (see e.g. [23] resp. [4])

LEMMA 3

(i) (R, P_{n}, 1) ≈ (M_{p}).

(ii) (R, exp{Φ(n)}, 1) ≈ R_{ϕ}, with Φ(x) := ∫_{1}^{x} d t/ϕ(t) (x ≥ 1).

Notation: For two summability methods (A), (B) : (A) ≈ (B) means s_{n} → s(A) ⇒ s_{n} → s(B) and vice versa.

In our context we are interested in Riesz-means with the sequences (λ_{n}) of special type, namely

λ_{n} = exp \{∫_{1}^{n} \frac{1}{l(t)}d t\}

with l(t) satisfying the conditions in (1.5). The second theorem of consistency, see [14, Theorem 19], easily implies the following equivalences:

LEMMA 4

(i) (R, log(n + 1), 1) ≈ (R, exp \{ ∫_{1}^{n+1}\frac{1}{x log x}dx\} , 1),

(ii) (R, n, 1) ≈ (R, exp \{∫_{1}^{n} \frac{1}{x}dx\} , 1),

Remark 3 Combining the above Lemmas, we get connections between (M_{p}) -means and Riesz-means with special (λ_{n}) sequences.

We now prove abelian results relating the above methods with (P) -methods.

PROPOSITION 1

(i) Let (p_{n}) be a sequence of reals satisfying (1.1) and (1.3A), then we have for the corresponding power-series method (P) and the (M_{p}) mean s_{n} → s(M_{p}) implies s_{n} → s(P) .

(ii) Let (P) be a power series method satisfying (1.5) with l(t) = o (t) , then

(3.4) \frac{1}{xl(n)} ∑_{n≤k≤n+xl(n)} s_{k} → s, (n → ∞), ∀x > 0 implies s_{n} → s(P).

Proof:

(i) see Ishiguro [15]. (ii) Tenenbaum [26, Lemma 2] gives the above result for the Borel-method, e.g. l(t) = \sqrt{t}. His proof uses heavily the special asymptotic properties of the weights p_{n} = 1/n! . Since our restriction (1.5) together with (1.5′) and l(t) = o (t) basically imply the same asymptotic properties for the weights, the main steps of his proof extend and we give only a short outline. Because of the regularity of the methods, we can assume s = 0 . As a first step we use Theorem 1 in Bingham/Goldie [3] to get (3.4) uniformly on compact x -sets in (0, ∞) . Then we define a nondecreasing, differentiable function h : [0, ∞) → R I such that h(n) = log t_{n}, n ∈ N I , with t_{n} as in Lemma 1, hence h(t) → log R, (t → ∞) . With h we carry out a parameter transformation and set

c_{n}(t) = \frac{p_{n} exp\{nh(t)\}}{p(exp\{h(t)\})}, n = 0, 1, 2 …; t > 0.

This transformation was introduced in the context of Tauberian theorems, see [18].

The properties of Δ_{n} for power series methods with weights according to (1.5’) imply:

(h′(t))^{− \frac{1}{2}} ∼ l(t) (t → ∞).

Now we can use the basic inequality in Lemma 1(ii) and the above asymptotic relation for l(.) to obtain through an application of the mean-value theorem that

(3.5) c_{n}(t) = O (\frac{1}{l(t)}) for |n − t| = O (l(t)) .

Introducing a further function ϵ(t) , see [7,21,22], with

\frac{ϵ(t)}{l(t)} → ∞, \frac{ϵ(t)}{t} → 0, (t → ∞),

we can split ∑_{n=0}^{∞} s_{n}c_{n}(t) in five parts, namely (writing m for [t] ):

\begin{aligned}∑_{n=0}^{∞} s_{n}c_{n}(t) & = \{ ∑_{n=0}^{m−ϵ(m)−1} + ∑_{n=m−ϵ(m)}^{m−l(m)} + ∑_{n=m−l(m)+1}^{m+l(m)−1} + ∑_{n=m+l(m)}^{m+ϵ(m)} + ∑_{n=m+ϵ(m)+1}^{∞}\} s_{n}c_{n}(t) \\ & = ∑_{1} + ∑_{2} + ∑_{3} + ∑_{4} + ∑_{5} . \end{aligned}

Using Abel’s partial summation, the uniformity in (3.4) and the basic growth condition in (3.5), we obtain by the monotonicity of c_{n}(t) in n for fixed t:

sup_{|n−m|= O(l(m))} |c_{n+1}(t) − c_{n}(t)| = O (\frac{1}{l(m)}) o (l(m)h′(m)) = O (\frac{1}{(l(m))^{2}} ) .

Now we obtain by Abel’s partial summation ∑_{3} = o (1) . That ∑_{1} + ∑_{5} = o (1)

follows again via Abel’s partial summation and the fact that ∑_{k=0}^{n} s_{k} = o (n) . Use

l(n) = o (n) and (3.4) to obtain this.

To get ∑_{2} + ∑_{4} = o (1) , we use the basic property |1 − \frac{l(n)}{l(m)}| ≤ 3\frac{ϵ(m)}{m} for

|m − n| ≤ ϵ(m) , compare [7,18,21,22,23].

Remark 4 Combining Propositon 1 (ii) and Remark 3, we obtain implications from (M_{q}) -methods to (P) -methods.

4 Strong Laws for the auxiliary summability methods

In this paragraph we formulate and extend strong law type results for the moving average methods and the (M_{p}) -means. These results are the basic tool for the proof of the first implication in our Theorem. We first need some notation:

(4.1) \{ \begin{matrix}S̄_{n,l(n)} = max_{0≤j≤l(n)} ∑_{k=n}^{n+j} X_{k} and \\ S_{n,l(n)}^{∗} = max_{0≤j≤l(n)} |∑_{k=n}^{n+j} X_{k}| \end{matrix}

Then we have

PROPOSITION 2

Let l(t) be a length-function as in (1.5), with l(t) = o (t) . Assume E(l^{−1}(|X|)) < ∞ and E(X) = 0 . Then we have

(i) lim sup_{n→∞} \frac{1}{l(n)}S̄_{n,l(n)} = 0 a.s.;

(ii) lim sup_{n→∞} \frac{1}{l(n)}S_{n,l(n)}^{∗} = 0 a.s.

Proof:

(i) We can follow the lines of the proof of Theorem 1 in [8]. There the case l(t) = t^{α}, 0 < α < 1 is treated. Since we have l(t) is self-neglecting, we only have to use the special properties of self-neglecting functions to adjust the arguments, see [4] resp.[17] for the details.

(ii) Use the sequences (X_{n}) and (−X_{n}) , then (i) implies (ii).

For convenience we add a result basically proved in [9], §5.2, Theorem 3.

LEMMA 5

Let (p_{n}) a sequence of reals satisfying (1.1) and (1.3A) and (p_{n_{ν}}) the nonzero sub-

sequence. Assume furthermore that the sequence 0 < \frac{P_{n_{ν}}}{p_{n_{ν}}} is strictly increasing

and that

(4.2) (\frac{P_{n_{ν}}}{p_{n_{ν}}} )^{2} ∑_{k=ν}^{∞} (\frac{p_{n_{k}}}{P_{n_{k}}} )^{2} = O (ν) .

Then we have

∑_{n=0}^{∞} P (p_{n}|X| > P_{n}) < ∞ ⇒ \frac{1}{P_{n}} ∑_{k=0}^{n} p_{k}(X_{k} − E(XI_{[p_{k}|X_{k}|≤P_{k}]})) → 0 a.s., that is

(X_{n} − E(XI_{[p_{n}|X_{n}|≤P_{n}]})) → 0(M_{p}),

with I the indicator function.

5 Proof of the main theorem

Before we prove the Theorem we need as a final preparation a Proposition, useful in its own right, which investigates the relation between (P) - convergence and moment conditions.

PROPOSITION 3

Let (P) be a regular power series method with a sequence of weights (p_{n}) , satisfying (1.1) and (1.5). Let as usual (X_{n}) be a sequence of i.i.d. random variables and (C_{n}) a sequence of real numbers.

Then (X_{n} − C_{n}) → 0(P) implies ∑_{n=0}^{∞} P (p_{n}|X_{n}| > Δ_{n}) < ∞.

Proof: Let (X′_{n}) and (X_{n}) be i.i.d. random variables and X_{n}^{s} = X_{n} − X′_{n}, then we have X_{n}^{s} → 0(P) . With Δ_{m} = p(t_{m})t_{m}^{−m}, see (2.1), we define

Y_{m} = ∑_{n=0}^{m} \frac{p_{n}t_{m}^{n}}{p(t_{m})}X_{n}^{s} and Z_{m} = ∑_{n=m+1}^{∞} \frac{p_{n}t_{m}^{n}}{p(t_{m})}X_{n}^{s}

for m = 0, 1, . . .. It follows that (Y_{1}, Y_{2}, … Y_{m}) and Z_{m} are independent for all m, and Y_{m} and Z_{m} are symmetrical. By Lévy’s second inequality and with Y_{m} + Z_{m} →^{p} 0, we get Z_{m} →^{p} 0. We now use Lemma 4, p.70 in [9], to obtain Y_{m} →^{a.s.} 0. Repeating the argument for Y_{m−1} and Z_{m} = Y_{m} − Y_{m−1} leads to

\frac{p_{m}t_{m}^{m}}{p(t_{m})}X_{m}^{s} = \frac{p_{m}}{Δ_{m}} X_{m}^{s} →^{a.s.} 0.

An easy application of the Borel–Cantelli–Lemma and the symmetrisation inequality [11, p.102] results in

\frac{p_{m}}{Δ_{m}} (X_{m} − m(X)) →^{a.s.} 0

with m(X) denoting the median of X . Since p_{m} = o (Δ_{m}) is a consequence of (1.5) the proof is completed.

Proof of the Theorem: We distinguish two cases.

(a) n = O (l(n)) Here we use as weaker summability method the corresponding (M_{p}) -means. Since

(a) implies \frac{np_{n}}{Δ_{n}} = O (1) the P_{n} are dominatedly varying and we have P_{n} ≤ Δ_{n} =

O (P_{n}) by Lemma 2. Therefore

E(l^{−1}(|X|)) < ∞ ⇔ ∑_{n=0}^{∞} P (p_{n}|X_{n}| > Δ_{n}) < ∞ ⇔ ∑_{n=0}^{∞} P (p_{n}|X_{n}| > P_{n}) < ∞,

and we can use Lemma 5 to obtain the implication from (i) to (ii), i.e

X_{n} − C_{n} → 0(M_{p}) a.s.

with C_{n} = E(XI_{[p_{n}|X_{n}|<Δ_{n}]}) or if E(|X|) < ∞, C_{n} = E(X) . Now Ishiguro’s Theorem, Proposition 1(i), implies

X_{n} − C_{n} → 0(P) a.s.

which is (iii). The concluding implication from (iii) back to (i) follows by Proposition 3.

(b) l(n) = o (n) . Now we use as weaker summability method the moving averages as defined in (3.4). With Proposition 2 we get the implication from E(l^{−1}(|X|)) < ∞ to convergence in (3.4), while Proposition 1 gives the convergence in the (P) -sense, i.e. X_{n} − E(X_{n}) → 0(P) a.s. As above Proposition 3 concludes the proof.

Restricting the class of power series methods we get the following

Corollary: Assume

(P) ∈ {(L_{α}), (A_{α}), (A_{R_{α}}), (A_{p}), (B(α, β)), (B_{p})} .

Then the following are equivalent

(i) E(l^{−1}(|X|)) < ∞

(ii) X_{n} − C_{n} → 0 (R, exp \{∫_{1}^{n} \frac{1}{l(t)}dt\} , 1) a.s.

(iii) X_{n} − C_{n} → 0(P) a.s with C_{n} = E(XI_{[p_{n}|X_{n}|<Δ_{n}]}) or if E(|X|) < ∞, C_{n} = E(X) .

Proof: We only have to show the equivalence of the summability methods used for (ii). We consider three cases.

(a) (P) ∈ (L_{α}) : We have n = O (l(n)) = O (n log n) . In the proof of Theorem 3 we used the corresponding (M_{p}) -means. Since all logarithmic-type (M_{p}) - means are equivalent to the logarithmic mean with p_{n} = 1/n, use [13] Theorem 14, we obtain our result by Lemma 3(i) and Lemma 4(i).

(b) (P) ∈ {(A_{α}), (A_{R_{α}})}: Since now n = O (l(n)) = O (n) again the corresponding (M_{p}) -means have been used. Here all (M_{p}) - means are equivalent to the Cesàro mean with p_{n} = 1 , again by Theorem 14 in [13]. Another application of Lemma 3(i) and Lemma 4(ii) gives the remaining equivalences.

(c) (P) ∈ {(A_{p}), (B(α, β)), (B_{p})}: Since now l(n) = o (n) the moving averages have been used in Theorem 3. But these are equivalent to the given Riesz-means by Lemma 3(ii).

Remark 5:

(i) In the above Corollary the (M_{p}) -means can also be used as auxiliary methods (see Remark 3).

(ii) We give a survey of our results in terms of moment conditions, the given moment condition being equivalent to convergence in the sense of the corresponding summability method.

( L_{α}) E (\frac{|X|}{log^{+} |X|}) < ∞. Here we have to adjust the X_{n} by subtrac-

ting C_{n} = E (XI_{[|X|< \frac{n}{α} log n]}) .

(A_{α}), (A_{R_{α}}) E(|X|) < ∞;

(A_{p}) E (|X|^{\frac{2}{2−α}} ) < ∞, p_{n} = exp\{n^{α}\}, 0 < α < 1 ;

(B(α, β)) E (|X|^{2}) < ∞;

(B_{p}) E (|X|^{\frac{2}{2−β}} ) < ∞, p_{n} = exp\{−n^{β}\}, 1 < β < 2 .

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