COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF ARRAYS OF RANDOM ELEMENTS

Let {Xnk:k,n=1,2,…} be an array of row-wise independent random elements in a separable Banach space. Let {ank:k,n=1,2,…} be an array of real numbers such that ∑k=1∞|ank|≤1 and ∑n=1∞exp(−α/An)l∞ for each α ϵ R


INTRODUCTION AND PRELIMINARIES.
Wei and Taylor [i] obtained the convergence of [k=l ank Xk in probability and with probability one by assuming tightness (given e > 0 there exists a compact set K such that sp P[X K e and uniformly bounded pth (p > i) moment conditions on n E the sequence of random elements {X in separable Banach spaces.Howell and Taylor n [2] proved Marcinkiewicz-Zygmund type weak laws of large numbers for the weighted sum V n { Xnk kk=l ank Xnk of arrays of random elements in Banach Spaces satisfying cer- tain geometric conditions.In this paper a stronger mode of convergence, complete convergence, is obtained for Lk=l ank Xnk in separable Banach spaces with less re- strictive conditions than the results of Wei and Taylor and without assuming geometric R.L. TAYLOR conditions on the spaces.The format for these results was motivated in part by the form of the kernel density estimates (which are weighted sums of arrays of random elements) and by the relative absence of laws of large numbers for arrays of random elements.
Let E denote a real separable Banach space with norm If Let (, A, P) denote a probability space.A random element X in E is a function from into E which is A-measurable with respect to the Borel subsets of E. The expected value of X is defined to be the Pettis integral (when it exists, see pp. 38-41 of [3]) and is th denoted by EX.The p moment of a random element X is E( IxI p) where E is the ex- pected value of the (real-valued) random variable lXl p.The concepts of independ- ence and identical distributions (i.i.d.) have direct extensions to E. A sequence of random elements {X is said to converge completely to the random element X if , complete convergence implies convergence with probability one, but the reverse implication need not hold.Because of the methods of proof, complete convergence will be obtained for the results of this paper rather than (the standard) convergence with probability one.

2.
The Complete Convergence of Weighted Sums.
In this section complete convergence for weighted sums of arrays of random elements is obtained.Since no geometric conditions are assumed on the space, it is easy to show that moments conditions alone will not suffice.Thus, sigma compact support or uniform compact integral conditions will be assumed on the distributions of the random elements.Throughout this section {Xnk} will denote an array of random elements in a separable Banach space E such that {Xnk: k e I} are independent for each n.Moreover, {auk} will always denote an array of real numbers such that k=l lank I and n=l exp[-/An < (2.I) for each > 0 where A 2 n k=l auk" First, the complete convergence of the weighted sums will be obtained when the random elements are restricted to a compact subset with probability one.This result will allow consideration of the random elements which are truncated to compact subsets and the corresponding parts off the compact subsets.THOREM I. Let K be a compact subset of a separable Banach space E. If {Xnk} is an array of random elements in E such that [Xnk: k -> i} are independent for each n and such that P[Xnk e K] i and EXnk 0 for all n and k, then, I.k=l ank Xnkll 0 completely.
PROOF.It can be assumed that K is convex and symmetric and 0 K (wlog).In the dual space E there is a countable set S {f with If I[ 1 which separate i i points of K. Also, for each e > 0 there exists {fl 'fm S such that m {x e K: II x II > E} u {x e K: [fi(x)I > }.
/// Using truncation to a compact subset and Theorem i, a strong law of large numbers will be proved for triangular arrays of random elements.
THEOREM 2. Let {Xnk} be an array of random elements in a separable Banach space E satisfying R.L. TAYLOR (i) {Xnk k > i} are i.i d for each n, (ii) EXnk 0 for all n and k, (iii) 'n=l E(IIXnl II2q)/nq < for some q 2 i, and (iv) given e > 0 there exists a compact subset K such that lim sup E(IIXnlII (2.7) Using ank in for k i,..., n and ank 0 for k > n, it follows that An I and 'n-lexp[-<I/An 'n-l-exp[-an] < oo.

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The form of Theorem 2 is very appropriate for applicat+/-ons in kernel density estimation where X I, ...,Xn is a random sample from a distribution with unknown probability density function f.The kernel estimate for f is given by t-X.
where K is an arbitrary kernel function which is usually assumed to be bounded, nonnegative and integrable to i and where h is a sequence of positive numbers tending n to zero which adjusts the height and spread of the kernel function.The random elements {Xnk} in an appropriate Banach subspace of function space can be defined as Xnk K(-----) --E K( h )" Conditions (i) and (ii) of Theorem 2 are trivially n n n n satisfied.For complete convergence in the Ll-norm, (iii) is easily satisfied since t-X I t-X I E(F_o I-- In the sup-norm, E IIXnlIl2q _< [_2 bound (K)]2q n .oon hn2 and thus )-q < n=l q > 1 is sufficient for (iii).Finally, when h 0, for some is achieved with traditional techniques, and (complete convergence) consistency of the kernel density estimate in the Ll-norm or the sup-norm is obtained from Theorem 2.
The verification of (iv) depends on the particular Banach space In partic- th ular, (iv) is implied by tightness and uniformly bounded r moments (r > i).It is th easy to see that tightness and uniformly bounded r moments (r > i) are not necessary for Condition (iv) of Theorem 2 even when E R.
The complete convergence for weighted sums of arrays of random elements will be obtained in Theorem 3.Only the parts of the proof which differ from the proof of Theorem 2 will be presented.First, let X denote the essential supremum of a non- negative random variable X.
THEOREM 3. Let {Xnk} be an array of random elements in a separable Banach space E satisfying (i) [Xnk k >-i} are independent for each n, (ii) EXnk 0 for all n and k, (2.13) completely by Theorem 2 of Chow [4].Thus, the proof is completed by following the same steps of the proof of Theorem 2.
/// Since sup I IXnkll < oo, tightness is sufficient for Condition (iii) of k,n Theorem 2. However, since lim sup .k=iIankl 0is also sufficient, tightness is not n necessary.Also, the more restrictive condition (iii) in Theorem 3 (than condition (iii) of Theorem 2) allows the use of more general weights {ank}.
In this section the complete convergence of weighted sums will be established for random elements in a Banach space which has a Schauder basis.Since the con- sistency of the kernel density estimates depend on a particular space which may have a basis, more general results can be obtained.The hypotheses will be shown to be more applicable when the Baach space has a Schauder basis.In particular, only coordinate-wise independence for some Schauder basis will be needed, and the weights .ti=lfi(x)bi and Qt(x) i;t+l fi(x)bi for each t 1,2,3,...It can be assumed without loss of generality that the basis is monotone (by renorming the space if necessary).
for each i, fi(Xnk): k > l} are independent, sub-Gaussian random variables for each n-with uniformly bounded parameters T(fi(Xnk)) (ii) EXnk 0 for each n and k, and (iii) lira .upB Q t(.k=l ank Xnk) ll 0, t-o n then k=l ank XnII 0 completely.PROOF.Given g > 0 pick t so that sup IZk=I ank Qt(Xnk) II < 2 (3.2) with probability one.Thus, for each n nce, from (i), (ii), (3.1) ,and (3.3) /// Theorem 4 allows for more general weights since Zk:l lankl bn need not be bounded.Note that in this case Zk=l ank Xnk is assumed to be a.s.convergent.Con- dition (i) is easily satisfied in general.For example, if P[Xnk E K] i for each n and k where K is a compact set, then {fi(Xnk)} are sub-Gaussian with parameter (fi(Xnk <-21/211fill (supl Ixl I).Also, by eemma 1.3.3 of Taylor [3] xK lim sup I IQ(Xnk) II 0 when P[Xnk E K] i.Thus, Condition (iii) also holds in this t n,k case when Zk=l lankl < i.Finally, as an aside, the conclusion of Theorem 4 hold.
almost surely (rather than completely) if the uniform almost sure convergence of Condition (iii) is replaced by simply almost sure convergence.
Coordinate-wise sub-Gaussian random elements are also appealing for applications since coordinate-wise independence suffices in this setting.In general, condition (iii) is the troublesome requirement which must be satisfied.The following lemma (a modification of Lemma 8.1 from Zaanen [6])allows (iii) to be replaced by a more restrictive condition but one which may be easier to verify in applications.
LEMMA.Let {Y t,n i, 2 be real-valued random variables satisfying tn (i) Ytn -> 0 with probability one for all t and n, (ii) for fixed n,Y(t+l)n -< Ytn with probability one for each lk=l ank Xnkll 0 a.s.PROOF.Condition (iv) implies that k=l.ank Xnk is defined with probability one.With probability one, sup Ik=l ank Xnkll can be identified as Y in the pre- n ceding lemma.Since E has a monotone basis, Ytn I]Qt(.k=lank Xnk) II satisfies (ii) of the lemma.Next let Y be identified as sup If0 t(k=l ank Xnk) II with prob-

III
The comparisons of general Banach space results with the results of this section will be completed by showing that condition (iv) of Theorem 3 implies condition (ii) of Theorem 5. Assume that .k=iIankl<-I for each n and that for compact, convex, symmetric set K can be chosen so that 0 e K and sup k=l lankl EFllXnklll[Xnk K] < !2. (3.4) n By Lemma 1.3.3 of Taylor [3], choose t O so that for all t e t O sup llQt(x)ll < .Then for t >-  5) since liQt(x) II <-If x If for all t.Thus, the compact uniform integrability of (3.4) (and hence tightness with uniformly bounded pth (p > 1) moments) is sufficient for Condition (iii) of Theorem 5. Identical distributions and moments conditions will suffice for Condition (iii).However, while it is often true (in applications) that {Xnk: k >_ i} are independent and identically distributed for each n, identical distributions seldom hold for the array {Xnk}.
exists a compact set K such that Then lim sup "k=l ankIE(I Xnkl II[Xn k K ]) < g. n Ilk=Iank Xnkll 0 completely.OUTLINE OF PROOF.Steps (2.6), (2.7), and (2.8) in the proof of Theorem 2 remain the same.The random elements {l[Znkll El [Znk[I} have zero means and are bounded by r with probability one.Hence, {IIznkll E IIZnkl I} are sub-Gaussian with parameters < r2 I/2 and Ik=I ank(IlZnkll El IZnklI)] 0 {b.} denote a Schauder basis for E and let {f.} denote the corresponding i coordinate functionals.For notation convenience let U t (x)

THEOREM 4 .
Let {Xnk} be random elements in a Banach space which has a Schauder basis.If R.L. TAYLOR