STRONG LAWS AND SUMMABILITY FOR SEQUENCES OF ’ -MIXING RANDOM VARIABLES IN BANACH SPACES

In this note the almost sure convergence of stationary, ’ -mixing sequences of random variables with values in real, separable Banach spaces according to summability methods is linked to the ful(cid:12)llment of a certain integrability condition generalizing and extending the results for i.i.d. sequences. Furthermore we give via Baum-Katz type results an estimate for the rate of convergence in these laws.


Introduction and main result
Let (Ω, A, IP ) be a probability space rich enough so that all random variables used in the sequel can be defined on this space. If X 0 , X 1 , . . . is a sequence of independent, identically distributed (i.i.d.) real valued random variables, then the almost sure (a.s.) convergence of such a sequence according to certain summability methods is equivalent to the fulfillment of certain integrability conditions on X 0 , see e.g. [5,6,12,19,24]. Some of the above results have been extended to sequences of stationary, ϕ-mixing sequences of real-valued random variables [3,5,22,28] and to i.i.d. Banach space-valued random variables [4,9,14,15,18,25]. The aim of this paper is to prove a general result linking convergence according to summability methods with integrability conditions for stationary, ϕ-mixing random variables taking values in a real separable Banach space (IB, . ) equipped with its Borel σ-algebra. A result of this type has useful statistical applications, e.g. to the empirical distribution function, to the likelihood functions, to strong convergence of density estimators and to least square regression with fixed design [25,27,31].
Before we state our main result we give the minimal amount of necessary definitions. We start out with the relevant summability methods ( for general information on summability see [13,29,32]). Let (p n ) be a sequence of real numbers such that (1.1)        p 0 > 0, p n ≥ 0, n = 1, 2, . . ., and the power series p(t) := ∞ n=0 p n t n has radius of convergence R ∈ (0, ∞]. We say that a sequence (s n ) is summable to s by the power series method (P ), briefly s n p n t n converges for |t| < R and σ p (t) = p s (t) p(t) → s for t → R − .
Observe that the classical Abel (p n ≡ 1) and Borel methods (p n = 1/n!) are in the class of power series methods. We assume throughout the following regularity condition with a real-valued function g(.), which has the following properties, see [8], is positive and non-increasing with lim t→∞ g (t) = 0; As the corresponding family of matrix methods we use the generalized Nörlund methods, see [20] for a general discussion. We say a sequence (s n ) is summable to s by the generalized Nörlund method (N, p * κ , p), briefly s n → s (N, p * κ , p), if where we define the convolution of a sequence (p n ) by p * 1 n := p n and p * κ n := n ν=0 p * (κ−1) n−ν p ν , for κ = 2, 3, . . . .
In order to define two more especially in probability theory widely used summability methods, we need a few function classes. We call a measurable function f : (0, ∞) → (0, ∞) We write briefly: f ∈ SN.
(iv) regular varying with index ρ, if We write briefly: f ∈ R ρ .
with φ ∈ SN . The above convergence is locally uniform in u (N.H.Bingham et al., [7], §2.11). Finally we call a sequence (s n ) summable by the generalized Valiron method (V φ ), briefly For a general discussion of these methods and a general equivalence Theorem see [20,30]. As a measure of dependence we use a strong mixing condition. We write F m n := σ(X k : n ≤ k ≤ m) for the canonical σ-algebra generated by X n , . . ., X m and define the ϕ− mixing coefficient by We say, that a sequence X 0 , X 1 , . . . is ϕ-mixing, if ϕ n → 0 for n → ∞.
(iii) For the case of real-valued random variables the Theorem was proved in [22] using techniques from [3,5,28].
(iv) The case of Abel's method, p n ≡ 1, is not directly included, but it can be viewed as a limiting case, compare [8]. For the real-valued mixing case the equivalence of (M ) ⇔ (S3) ⇔ (S4) has basically been proved in Theorem 6 in [3] and in the i.i.d. Banach-valued case in [9]. Observe that the matrix method in (S3) is Cesàro's method.
(v) Let Y 1 , Y 2 , . . . be ϕ-mixing and uniformly distributed on (0, 1) and F n (t) = n −1 n i=1 1 [Yi≤t] be the empirical distribution function based on Y 1 , Y 2 , . . . Y n . As in Lai [25] the above theorem might be extended to discuss the specific behaviour of F n (t) − t for t near 0 and 1. Likewise one can use the theorem to discuss certain likelihood functions (see also [25]).
(vi) The above theorem can also be used to obtain results on strong convergence of kernel estimators in non-parametric statistics in the spirit of Liebscher [27] (see also [23] for results of Erdős-Rényi-Shepp type related to kernel estimators).
(vii) Consider the problem of least square regression with fixed design (for a precise formulation of the problem and background see [31], §3.4.3.1). In that context stochastic processes of the form {n − 1 2 n i=1 θ(x i )e i : θ ∈ Θ} play a central role (typically x i ∈ IR d , Θ is a set of functions with θ : IR d → IR and the error terms e i are i.i.d.). Imposing regularity conditions on Θ the theorem can be used to discuss the speed of convergence of the above process (even if independence is replaced by the appropriate ϕ-mixing condition).

Auxiliary results
We start with an application of the Feller-Chung lemma for events generalizing a result for i.i.d. real-valued random variables in [12], Lemma 2.
→ 0. The proof follows the lines of Lemma 2 in [12] using only standard properties of ϕ-mixing sequences, e.g. Lemma 1.1.1 in [17]. As a key result we now prove a Lévy-type inequality using techniques from [25], Lemma 1, {X n } be an independent copy of {X n } and consider the symmetrized sequence {X s n }, such that for every n X s n = X n − X n . Denote by S s n = n k=1 X s k . Then we have for every ε > 0 Proof: According to Bradley [10] Theorem 3.2 the ϕ-mixing coefficients for {X s n }, which we denote by {ϕ s n }, cannot exceed twice the size of the ϕ-mixing coefficients for {X n }. So we have ϕ s n ≤ 2ϕ n ∀n. Now we can basically follow the proof of Lemma 1 in [25] with the only modification being that we use the definition of {ϕ s n } instead of independence. We outline the main steps. For notational convenience we assume, that {X n } itself is the symmetrized sequence. First assume that X k = (X k . Then we claim: Define the stopping times with inf ∅ = n + 1. For k = 1, . . . , n consider the following sets Observe that for 1 ≤ k ≤ n we have A where we define k ∈ σ(X 1 , . . . X k ). Using the mixing condition (instead of independence as in Lai's Lemma) we get for k < n, Obviously With the same arguments we get for 1 ≤ j ≤ d, Turning to the general case we find, since IB is separable, a countable, dense subset D := for all b ∈ IB, see [16] p.34. Since {X n } is a symmetrized sequence, so is {f(X n )} for all f ∈ IB with ϕ-mixing coefficients not exceeding the mixing coefficients of {X n } (see [10], p.170). Therefore

3 A reduction principle
We now state and prove a general reduction principle which allows us to deduce results for IB-valued random variables from the corresponding results for real-valued random variables. Let (V ) be a summability method with weights c n (λ) ≥ 0, n = 0, 1, . . .; λ > 0 a discrete or continuous parameter We say s n → s (V ), if V s (λ) → s (λ → ∞). Assume that if {X n } is a stationary ϕ-mixing sequence of real-valued random variables with mixing coefficient ϕ 1 < 1/4 and if ψ is a function as in our Theorem, then IE(ψ(|X|)) < ∞, IE(X) = µ ⇒ V X (λ) → µ a.s..

Under these assumptions we have
Proposition. If {X n } is a stationary ϕ-mixing sequence of IB-valued random variables

Proof:
Since IB is separable, we can find a dense sequence (b n ), n ≥ 1. For each n ≥ 1 define Hence for each n (A ni ) ∞ i=1 is a partition of IB.
Since IE( X ) < ∞ we furthermore get Consider now the sequence 1 {bi} (σ (2) mn(Xk )) of 0 − 1-valued random variables. Using again [10], p.170 and [11], Proposition 6.6, p.105, we see that this sequence is also stationary and satisfies the mixing condition. This sequence converges in the (V )-sense almost sure to Since we have it furthermore follows, that σ (2) mn (X k ) is almost sure (V ) summable for each pair m, n ≥ 1 to Using the triangle inequality we get and V X(ω) (λ) is a Cauchy-sequence in IB. Since IB is complete (V )-summability of {X n } follows.
For simple random variables we immediately see from our proof, that the (V )-limit is the expected value. Using the approximating property of simple functions, see [16], pp.76-82, the same is true in the general case i.e. X n → µ = IE(X) (V ) a.s..
Since IE(ψ( X )) < ∞ and hence IE( X ) < ∞, we find for Consider the set A := {b 1 , b 2 , . . .b d } and define random variables with partial sums S n and S n . Now the X k assume only finitely many values and the sequence {X n } is stationary and satisfies the mixing condition. Furthermore IE(X ) = 0 and IE(ψ( X )) < ∞. Using the first part of the proof it follows that ∞ n=1 n −1 (ψ(n + 1) − ψ(n))IP max j≤n S k > εn < ∞ ∀ε > 0.
Furthermore { X n } is a stationary ϕ-mixing sequence of real-valued random variables with IE( X n ) < ε and IE(ψ( X n )) < ∞. Hence using again the Baum-Katz-type law in [22] for the partial sumsS n = n k=1 ( X k − IE( X k )).
Since X k = X k + X k we have S n = S n + S n and the claim follows.
Using the method of associated random variable (see [21] Lemma 3.2.2 or [28], proof of Theorem 1) we can assume, that the X s n are mutually independent and IE(ψ( X s )) < ∞ follows from [25], Theorem 1. Now (M ) follows as in the first proof. 2 Remark 4. For i.i.d. real-valued random variables our Theorem above is complemented by an Erdős-Rényi-Shepp type law, see [23], which is proved by using a result on large deviations of the above convergence. In the case of i.i.d. random variables taking values in a Banach space such a result is only known for the (C 1 )-method resp. (M φ )-method with φ(t) = t, see [1,2]. Hence in the case of our general summability methods this interesting question remains subject to further research.