STABILITY FOR RANDOMLY WEIGHTED SUMS OF RANDOM ELEMENTS

Let {Xn :n= 1,2,3, . . .} be a sequence of i.i.d. random elements taking values in a separable Banach space of type p and let {An,i : i= 1,2,3, . . . ;n= 1,2,3, . . .} be an array of random variables. In this paper, under various assumptions of {An,i}, the necessary and sufficient conditions for ∑∞ i=1An,i Xi → 0 a.s. are obtained. Also, the necessity of the assumptions of {An,i} is discussed.


Introduction.
Let {X n : n = 1, 2, 3,...} be a sequence of independent identically distributed (i.i.d.) random variables and let {a n,i : i = 1, 2,...,n; n = 1, 2, 3,...} be a triangular array of constants.Many papers were devoted to extending various types of convergence modes to weighted sums W n = n i=1 a n,i X i in the literature.However, we are only interested in the work of almost sure convergence.The sequence { n i=1 a n,i } converging to 0 at a certain rate as n → ∞ is a traditional assumption.For example, under the assumption n i=1 a 2  n,i = O(n −2/r ), W n → 0 a.s.if E|X 1 | r < ∞ and EX 1 = 0 (See Chow and Lia [3] and Choi and Sung [2]).On the other hand, Padgett and Taylor [4] extended the usual convergence theorems to weighted sums of random elements in a separable Banach space.It would be interesting to extend the results with random weights.
Let {A n,i : i = 1, 2,...,n; n = 1, 2, 3,...} be a triangular array of random variables such that n i=1 A 2  n,i = O(n −2/r ) a.s., Ahmad [1] obtained r < ∞ and EX 1 = 0. We note that, for the Marcinkiewicz-Zygmund law of large numbers, we take the uniform weight a n,i = n −1/r but the condition n i=1 a 2 n,i = O(n −2/r ) cannot be satisfied.The purpose of this paper is to extend the randomly weighted sums of a triangular array of random variables to that of an infinite array of random elements such that the Marcinkiewicz-Zygmund law of large numbers can be obtained as a corollary.
In Section 2, we establish the Marcinkiewicz-Zygmund law of large numbers in a separable Banach space of Type p.In Section 3, we consider an infinite array of random variables {A n,i : n, i = 1, 2, 3,...} as the weight under various assumptions of {A n,i }, we obtain that W n = ∞ i=1 A n,i X i → 0 a.s.if and only if EX 1 = 0 (when it exists) and E X 1 r < ∞.
2. The Marcinkiewicz law in a space of type p.Let (Ω,F,P) be a probability space and B be a real separable Banach space with norm • .A random element is defined to be an F -measurable mapping of Ω into B with the Borel σ -field.The concept of independent random elements is a direct extension of the concept of independent random variables.A detailed account of basic properties of random elements in real Banach spaces can be found in Taylor [6].
In this section, we prove the Marcinkiewicz-Zygmund law of large numbers in a space of type p.First, we introduce a space of type p. Definition 1.Let 1 ≤ p ≤ 2 and {r i : i = 1, 2, 3,...} be a sequence of independent random variables with Pr(r i = ±1) = 1/2.A separable Banach space B is said to be of type p if there exists a constant C such that for every n ∈ N and all x 1 ,...,x n ∈ B.
Woyczyński [7] proved the equivalent condition of a space of type p.
Lemma 1 (Woyczyński [7]).Let 1 ≤ p ≤ 2 and q ≥ 1.The following properties of B are equivalent : (i) The separable Banach space B is of type p. (ii) There exists C such that, for every n ∈ N and for any sequence (2.2) Using Lemma 1, some elementary properties of spaces of type p can be easily proved.Every separable Hilbert space and finite-dimensional Banach space are of type 2. Every separable Banach space is at least of type 1, and the p and L p are of type min{2,p} for p ≥ 1.If B is a space of type p and 1 ≤ q ≤ p, then B is a space of type q.Before considering the Marcinkiewicz-Zygmund law of large numbers in a space of type p, we need the following definition and lemmas.Definition 2. Let B be a separable Banach space, B * the dual space of B, and B the unit ball in B * .X is a random element in B. The directionally maximum median of X is defined by where µ(Y ) denotes the minimum median in absolute value of the random variable Y .
Lemma 2 (Sakhanenko [5]).Let X 1 ,...,X n be independent random elements in B and We obtain S n converges to some random element a.s., and S n converges to S in probability.Hence, S n converges to S a.s.Now, we prove the Marcinkiewicz-Zygmund law of large numbers in a space of type p. Theorem 1.Let B be a separable Banach space of type P and {X n : n = 1, 2, 3,...} be a sequence of independent and identically distributed random elements in B. Then, for any 0 < r < p, Proof.Necessary part.Since (2.8) holds, whence, by the Borel-Cantelli lemma, (2.10) (2.11) ) converges a.s.By Kronecker lemma, (S n −nc)/(n 1/r ) converges a.s.for any constant c.
3. The convergence of the weighted sums.Throughout this section, we deal with the almost sure convergence of randomly weighted sums ∞ i=1 A n,i X i , where {X n : n = 1, 2, 3,...} is a sequence of independent and identically distributed random elements in a space of type p and {A n,i : n, i = 1, 2, 3,...} is an array of random variables satisfying some conditions.Theorem 2. Let B be a separable Banach space of type p.Let {X n : n = 1, 2, 3,...} be a sequence of independent and identically distributed random elements in B such that E X 1 r < ∞ and r < p.Moreover, we assume that EX 1 = 0 when r ≥ 1.Let {A n,i : n, i = 1, 2, 3,...} be an array of random variables such that {A n,i } and {X i } are independent and satisfying for every n and some constant M > 0. (3.4) for all arrays {A n,i } satisfying the above conditions, then E X 1 r < ∞.
Proof.Since {A n,i } and {X i } are independent, if r > 1, we choose p = q = r in Lemma 1, then (3.9) Since {i 1/r A n,i } is bounded a.s.for every n and r , lim We have for all arrays {A n,i } satisfying the above conditions, we can choose Then (2.8) holds.By Theorem 1, we have E X 1 r < ∞.
Remark 1.The following example claims that condition (3.4) cannot be omitted.Consider the real number space R as a space of type 2. Choose a sequence {X n : n = 1, 2, 3,...} of independent and identically distributed random variables with EX 2  1 < ∞ and EX 1 = 0. Define There is a constant C > 0 such that n i=1 |a n,i − a n,i+1 | ≤ Cn −1/r .We have |a n,i | ≤ Cn −1/r for every i.So, lim n→∞ a n,i = 0 for every i. Therefore, So, the proof is complete.
The assumptions of {A n,i } in Theorem 2 can be simplified as in Theorem 3 for r < 1 and Theorem 4 for r ≥ 1.

Lemma 4. Let {b
If the result of this Lemma is false, then any n, l > 0, there exists i > l such that b i > n/i.We define And We see that When r < 1, Theorem 2 can be rewritten as follows.
Theorem 3. Let B be a separable Banach space of type p.Let {X n : n = 1, 2, 3,...} be a sequence of independent and identically distributed random elements in B such that E X 1 r < ∞ and r < 1.Let {A n,i : n, i = 1, 2, 3,...} be an array of random variables such that {A n,i } and {X i } are independent, and satisfying and for every n and some constant M > 0. (3.27) r ) a.s.for every n.From the proof of Theorem 2, we have lim i→∞ Y i = 0 a.s. and , where the definitions of D 0 , D 1 i , and D 2 i are the same as in Theorem 2. For every w ∈ D and every > 0, we can choose A such that Y i (w) < The proof of the converse part is the same as the proof of Theorem 2. So the proof is complete.
When r ≥ 1, we can obtain the following theorem: Theorem 4. Let B be a separable Banach space of type p.Let {X n : n = 1, 2, 3,...} be a sequence of independent and identically distributed random elements in B such that EX 1 = 0 and E X for all arrays {A n,i } satisfying the above conditions, then E X 1 r < ∞.
Proof.We see that

.36)
So, from the proofs of Theorem 2 and Theorem 3, we can obtain this theorem.Now, we consider a very special case of {A n,i }.Let A n,i = n −1/r for i = 1, 2,...,n and A n,i = 0 for i > n.The assumptions of {A n,i } in Theorem 4 can be easily verified.Therefore, the Marcinkiewicz-Zygmund law of large numbers in a space of type p can be obtained as the following corollary.
Corollary 2. Let B be a separable Banach space of type p and {X n : n = 1, 2, 3,...} be a sequence of independent and identically distributed random elements in B with zero means.For any and the proof is complete by Theorem 4.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

First
Round of ReviewsMay 1, 2009 |≤M a.s.for every n and lim n→∞ B n,i = 0 a.s.for every i.Define D 0 = {w : lim i→∞ Y i (w) = 0} and D 1 n = w : ).Hence, ∞ i=1 |B n,i i = {w : lim n→∞ B n,i (w) = 0} for each i.For every w 1 r < ∞ for 1 ≤ r < p.Let {A n,i : n, i = 1, 2, 3,...} be an array of random variables such that {A n,i } and {X i } are independent, and satisfying