STRONG CONSISTENCIES OF THE BOOTSTRAP MOMENTS

Let X be a real valued random variable with E | X | r + δ ∞ for some positive 
integer r and real number, δ , 0 δ ≤ r , and let { X , X 1 , X 2 , … } be a sequence of 
independent, identically distributed random variables. In this note, we prove that, 
for almost all w ∈ Ω , μ r ; n * ( w ) → μ r with probability 1 . if 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> lim n → ∞ inf m ( n ) n − β > 0 for 
some \tfrac{{r - \delta }} {{r + \delta }}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> β > r − δ r + δ , where μ r ; n * is the bootstrap r t h sample moment of the bootstrap sample some 
with sample size m ( n ) from the data set { X , X 1 , … , X n } and μ r is the r t h moment of 
 X . The results obtained here not only improve on those of Athreya [3] but also the 
proof is more elementary.

be the empirical distribution functions associated with the sequence {Xl{W ).X2{w ).X3(w ).For every positive integer n and w E f}, let {Xnl{W ).Xn2(W) XrunCn){W)} be independent, identically distributed random variables with distribution function Fn(X'W} defined as in (I.I}.We call {Xnl{W }.Xn2{W }. Xnm{n}{W}} the bootstrap sample set with bootstrap sample size m(n}; it is required that m(n} as th n .Denote by Pn.r{W} and n.r{W} the r sample moment of {Xl{W ).th

X2{w
Xn{W)} and the bootstrap r sample moment of {Xnl{X ).Xn2(w th Xnm(n{W)} respectively and denote by r the r moment of X. {When r=l, we use gn(W) and (w) instead of gn;l(W) and gn.l(W)" further gn{W) and gn(W) are called sample mean and bootstrap sample mean respectively.
A problem, from the bootstrap theory of Efron [1], is to find conditions such that, for almost all w, the bootstrap sample mean converges to the population mean {when it exists).That is, for almost all w.
ln.r{W) -*I r as n -m {1.2} with probability 1.By using the abstract "Vasserstein's metric" among distributions and a Rallow type inequality, Bickel and Freedman [2] showed that if EIX < w, then for almost all w E , {2) holds in probability.Athreya [3] found that if EIXI 0 < for some 0 1, and lim inf m{n)n-/3 > 0 for some /3 > 0 such that > I, then for almost all w e , {1.2) holds with probability 1.To show this he used the difficult and complex inequality of Kurtz [d].Bickel and Freedman and Athreya used deep   mathematics and hard inequalities to prove the consistency of the bootstrap sample mean to the population mean.Their proofs are not easily comprehended.This note, provides an elementary way to obtain the strong consistency, relying on the }4arkov inequality.oreover, the consistency property holds under weaker conditions than those presented in Athreya [3].
2. RESULTS AND PROOFS THEOREb[ 2.1.Let {X.X 1. X 2 be a sequence o[ tndependent, tdenttcallN dtstrtbuted random vartables vtth E[X[ r+i < for some integer r and real number 5 r.Then, for almost all w 9, (1.2) holds atth probabtlttN 1, tf lira inf n-o m(n)n-/3 > 0 for some real number / > 0 such that /3 > r-'" First. a lemma is needed in proving the theorem.The lemma is known in the literature.For the sake of completeness, a proof for the lemma is given.
2.2.Let {X, X 1. X 2 be a sequence of independent, identically distributed random variables.Then.for any 0 < p ( 1, EIX[ p ( implies that n=l n Thus, by the "three series Theorem" of Kolmogorov the lemma is established.
We are now in a position to prove the main result, which provides the strong consistency of the bootstrap sample moments.
PROOF OF THEORF 2.1.It suffices to prove the result for the case r=l.The other cases can be proved in a similar way with minor changes.Recall that, for each n and w E f}. {Xnl{W), Xn2{w XnmCn){W)) are the independent, identically distributed random variables with distribution function defined in {1.1).By the strong law of large numbers, we have for almost all w /n{W) -H as n --. (2.1) Thus. it suffices to show that.for almost all w I:{w} .n{W}l--... o as with probability I. From the Borel-Cantelll lemma, we only need to prove for almost all w I and for every e > O, r.
I.-2-i=l Xni (w) llnCW > e Ix c-).x2c.) x.c.) < (R). n=1 For the case of presentation, we suppress all the symbol w in Xi(w ).Xni(W) and ttn(W 1-8 and the symbol n in re(n).Let q be an integer such that 1 ( -1-and from the q Xarkov inequal ty, we have where the third equality in (2.4) holds since Xnl, Xn2 Xnmare identically distributed.Further the last equality in (2.4) is Justified since X Xn2,.. X nl nm are independent and E{Xnl-n 0 implies that there is no contribution for those terms which contain at least one of qi=l.In the sequel, we use the shorthand since qj 2. ql+q2+...+qt 2q and where the first inequality in (2.5) is obtained for fact that [a+bl s 2S(lalS+]b[ s) for a. b and s real numbers.Since I is finite it follows from (2.1) that.for almost all w.there exists a constant C such that /a n < C for every n.Further note that 5 < and qj 2 for J 1.2 q which imply > I. Thus.for almost all w qj qj-l+5 qJ q.