ON COMPLETE CONVERGENCE FOR Lp-MIXINGALES

We provide in this paper sufficient conditions for the complete convergence for the partial sums and the random selected partial sums of B-valued Lp-mixingales.


Introduction and results. McLeish
introduced first the concept of mixingales, a generalization of the concepts of mixing sequences and martingale differences, where the mixingale convergence theorems and the strong laws of large numbers have been proved.Furthermore, McLeish [6,8] studied the invariance principles for mixingales.Yin [9] generalized McLeish's concept of mixingales to operator-valued mixingales, and proved the operator-valued mixingale convergence theorems.Hall and Heyde [2] also pointed out that mixingales include martingale differences, lacunary functions, linear processes, and uniformly mixing processes (also called Φ-mixing).
On the other hand, up till now, there have been an extensive literatures in complete convergence for independent and dependent random sequences (especially, martingale differences and various mixing sequences), see partially the references here.However, there are few papers reported on the complete convergence for mixingales; see, for example, Liang and Ren [5].
Preceding observations stir us to investigate the complete convergence for mixingales.In the present paper, we first generalize slightly McLeish's definition of mixingales to B-valued L p -mixingales, and then give some general results about complete convergence for B-valued L p -mixingales.
Next, we introduce some notations.Let (B, • ) be a Banach space.B is said to be q-smooth (1 ≤ q ≤ 2) if there exists a constant C q > 0 such that for every B-valued random variables on a probability space (Ω, Ᏺ,P), and let {Ᏺ n ; −∞ < n < ∞} be an increasing sequence of sub σ -fields of Ᏺ.Then {X n , Ᏺ n } is called a L p -mixingale if there exist sequences of nonnegative constants C n and ψ(n), where ψ(m) ↓ 0 as m → ∞, which satisfy the following properties: for all n ≥ 1 and m ≥ 0, where X p = (E X p ) 1/p .Let {X n ; n ≥ 1} be B-valued random variables, and X 0 be a real nonnegative random variable.We call that {X n } is bounded in probability by X 0 (abbreviated Given a positive function l(x) defined on (0, +∞), we say that l(x) is a slowly variable function as x → ∞, if for all c > 0, see also Laha and Rohatgi [4].
From now on, we use C to denote finite positive constants whose value may change from statement to statement.For real numbers x, y, [x] denotes the largest integer k ≤ x, and x ∧ y means min(x, y).
The following are the main results of this paper.
Based on Theorem 1.2, we can now obtain the analogue to random selected partial sums of L p -mixingales.
and X is a real nonnegative random variable satisfying {X n } < X. Suppose {ν n ; n ≥ 1} are random variables which only take positive integer values and are defined on the same probability space as {X n }.Suppose l(x) is an increasing slowly variable function as x → ∞.If E X t+δ l(X t+δ ) ln + X < ∞, and there exist positive constants λ, β, and η such that (1.10) Remark 1.4.To our best knowledge, even if B = R (the real numbers), the results here are new.Furthermore, conditions (1.4), (1.6), and (1.9) are reasonable.For this purpose, we now particularize the general situation as follows.Let B = R.In return, q = 2. Let t = 1, p = 2 and {X n , Ᏺ n } be a L 2 -mixingale (coinciding with mixingale of McLeish [7] or Hall and Heyde [2]).Consequently, 1 ∧ 3(q/t − 1)

Remark 1.5. In general, if {C
In any case, roughly speaking, conditions such as plus a specific rate of convergence of ψ(m) to 0, ensure conditions (1.4), (1.6), and (1.9).Remark 1.6.Condition (1.8) is just one which is usually employed in literatures.

Proofs of the main results.
For the sake of convenience, we begin with two lemmas, which will be needed below.
We refer to Bai and Su [1] and Hu [3] for a proof of Lemma 2.1.By applying integration by parts, it is easy to prove the following lemma.Proof of Theorem 1.1.We write α = 1/t.Notice first that (2.2) By Chebyshev inequality, C r -inequality, Lemma 2.1, and L p -mixingale property we have which proves (2.4).Similarly, we obtain which is exactly (2.6).