797 Internat. J. Math. & Math. Sci. VOL. 14 NO. 4 (1991) 797-802 STRONG CONSISTENCIES OF THE BOOTSTRAP MOMENTS TIEN-CHUNG HU Department of Mathematics National Tsing Hua University Hsinchu, Taiwan 30043, R.O.C. and Department of Statistics University of North Carolina Chapel Hill, N C 27599 U.S.A. (Received November 14, 1990 and in revised form January 23, 1991) r+ < for some positive Let X be a real valued random variable with be a sequence of r, and let {X, X integer r and real number. 6. 0 < 5 I. X2 independent, identically distributed random variables. In this note, we prove that, for almost all w e with probability 1. if lim inf m > 0 for r.n(W) EIXI CF. , > r-5 (n)n- r r;n th sample moment of the bootstrap sample th with sample size m{n) from the data set {X I, X Xn} and is the r moment of 2 X. The results obtained here not only improve on those of Athreya [3] but also the some where is the bootstrap r Pr proof is more elementary. KEY WORDS AND PHRASES. Bootstrap sample probabi 1980 AMS size, Sample moment., Convergence with ty 1. SUBJF CLARIFICATION CODES. Primary 60F15; secondary 62C05. INTRODUION r+5 Let X be a real valued random variable with < for some positive real r, and let {X, X number 6 be a sequence of independent, identically I. X2 I. EIXI distributed random variables. Fn(X’W) n Let n 2 i=1 l[Xi(w x], n 1,2,..., w e , {I.I) T.C. HU 798 {Xl{W ). X2{w ). {Xnl{W ). Xn2(W) be the empirical distribution functions associated with the sequence ). X3(w XrunCn){W)} For every positive integer n and w E f}, let be independent, identically distributed random variables with distribution Fn(X’W} function . We call (I.I}. defined as in Xnm{n}{W}} {Xnl{W }. Xn2{W }. the bootstrap sample set with bootstrap sample size m(n}; it is required that m(n} th sample moment of {Xl{W ). the r and Denote by n n.r{W} Pn.r{W} X2{w Xn{W)} Xnm(n{W)} respectively and the bootstrap r gn(W) and (w) th gn;l(W) sample moment of r and denote by instead of and the r th gn.l(W)" [1], {Xnl{X ). Xn2(w moment of further sample mean and bootstrap sample mean respectively. theory of Efron as X. gn{W) {When r=l, we use and gn(W) are called A problem, from the bootstrap 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) with probability 1. -*I r as {1.2} n -m By using the abstract "Vasserstein’s metric" among distributions < w, then [2] showed that if EIX and a Rallow type inequality, Bickel and Freedman for almost all w E for some 0 , {2) holds m{n)n-/3 1, and lim inf almost all w e 0 Athreya [3] found that if < 0 for some /3 > 0 such that > I, then for EIXI in probability. > , {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 2. [3]. RESULTS AND PROOFS THEOREb[ 2.1. Let {X. X 1. X2 dtstrtbuted random vartables vtth 5 r. Then, for almost all w be a sequence o[ tndependent, E[X[ r+i < for some integer r (1.2) holds atth probabtlttN 9, tdenttcallN and real number 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 For the sake of completeness, a proof for the lemma is given. literature. Let {X, X 1. X be a sequence of independent, identically 2 distributed random variables. Then. for any 0 < p ( 1, EIX[ p ( implies that 2.2. 799 STRONG CONSISTENCIES OF THE BOOTSTRAP MOMENTS < Y. ty I. wi th probabi n=l PROOF. Let Y n Ixn n Ixn Thus lip . e(IXnl Ixn : P( ]x p > > n l/p) < n} n=l since EIxI p < -. Defining : E(IYn =) n=l Aj n=l J=l {{j-l) 1/p "fAJ a/p n (j)I/p}, < Ixl IX ladP j=l . ElY Choosing a I, .J’^j Ix = dP Ixl p c EIxl p < and 2, we have that <" and 2 Var Y n n=l q c J=l where the constant C depends only on a and p. n--1 :z n=J na/p j=l Ix we have. for a . < 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 are the independent, identically XnmCn){W)) 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) Thus. it suffices to show -H as n with probability I. (2.1) that. for almost all w I:{w} .n{W}l --... o all w --. as From the Borel-Cantelll lemma, we only need to prove for almost I and for every e > O, T.C. HU 800 r. Xni (w) llnCW > I.-2- i=l n=1 Ix c-). x2c.) e - For the case of presentation, we suppress all the symbol w in ( Let q be an integer such that 1 q and the symbol n in re(n). Xarkov inequal ty, we have m m P[] il Xni-ln] )e]XI’X2 Xn} x.c.) < (R). Xi(w ). Xni(W) and ttn(W 1-8 and from the 1- {_)2q E[{i=l{Xni_ln))2q[xi.x2 Xn] {2.3} Now we write n m . m m Y. E 12q=1 ii=1 i2=1 2q Y" t=l ql+...+qt=2q qil,i=l (Xni 2q-*n) IX (Xni l-ln) {2q)! Y . r qt"((7) ql E(Xnl-n C2q! qi22. i=l xn] Xl.X2 since Further the last equality in E{Xnl-n b for a n n n q m t=l t (2.4) is Justified . qi=l. In -.n c)t i-1- CXi-Xn) since identically Xnl Xn2,.. as n the sequel, we use the shorthand Thus, from [(Xnl-,n)ql x t:l O(bn) Xnmare Xnl, Xn2 0 implies that there is no contribution for those terms which contain at least one of notation a XnJ Xn] E (2.4) holds where the third equality in are independent and qt {Xnt-/n) IX ,x2 t ...E[{Xnt-.n}qt distributed. q Xnl ,..., {7) [{Xnl_btn}ql Ix.x. "+qt =2q ql +" 2 t q t=l ,X (x-.n) i--1 (Xi-n) (2.4) we have Xnm 801 STRONG CONSISTENCIES OF THE BOOTSTRAP MOMENTS n n q Ix+. lqt+ l..lqt)] 2q 2 qt (Ixi t+ i=l since 2q and where the first inequality in (2.5) is obtained 2. ql+q2+...+qt qj for fact that [a+bl s Inl qt) 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 Further note that 5 < and qj 2 for J q which imply /a < C for every n. 1.2 n > I. Thus. for almost all w - qj qj-l+5 lnl qJ < -. + i=l Now applying our lemma qj to t.2 qj-t++ < qt and choosing p (2.6) q. 1. we have. for almost all weft. .1.qj-l+5 i=1 Without loss of generality we put I-_._5 1+5 then from IX i iqj tT < m. J n/ where mCn) 1.2 {2.7) q. / is some real number such that {2 5} C2.6) and C2 7) we have 2qJg+{ l-/) t q Denote the exponent of @(t) (} in 2 + (1-/)t (2.8) as @(t) for t qj J=l qj-l+5 2 2- t(/-I + T- t t qJ J=l qj-l+5 1.2 q. Note that T.C. HU 802 1-5 (2.0) 1-6 -) q( 1,2 for q, where (2.9) holds since q-l/6 < 2 " for J 1,2 (2.2) follows From (2.3). (2.8) and (2.10). appropriate choice of q. By the t. This completes the proof. Theorem of Athreya. stated in Corollary 2.3. is an immediate consequence of our Theorem 2. I. 03ROSY 2.3. EIXI < and re(n) If E X2 < and If pn for some p ) 1, then (2.2) holds, for r=l. COROY 2.4. PROOF. . p > O. case By letting 6 m(n) for some p > O, then (2.2} holds for l-a for some p Let m(n) growth with an algebraic rate: that is, m(n) for some First note that 1+6 in our notation plays the role of 8 in Athreya’s. In I-6 < 8 < 2 we have > Hence, in this case, our condition Is strictly n 1-> -. weaker than the one is posed by Athreya. is even much weaker than if O ’ n p > In case 8 2. we only require ) 0 which the requirement in the Theorem 2 of Athreya. 1, then both of our theorem and Athreya’s theorem require > However, I. Recently. the author learned that Professor Csorgo also improved the result of Athreya using a different approaching. ACXNO. The author is most grateful to the Department of Statistics, University of North Carolina, Chapel Hill, for their hospitality. The author also wishes to thank the editor and a referee for helpful suggestions. I. Efron, B. Bootstrap methods: Another look at the Jacknlfe, Stattsttcs 7_. (1979). 1-26. .nnals o? Bickel. P.J. and Freedman. D.A. Some asymptotic theory for the bootstrap. _Ats of Stattsttcs 9_ (1981), 1196-1217. Athreya, K.B. Strong law for bootstrap, Statistics and Probablttto Letters L (.os3). v-5o. 4, Kurtz. T.G. Stctttsttcs Inequalities for laws of large numbers. A_n/re_is o/r lathemattcat 1874-1883. 43 {1972), Chung. K.L. A Course in Probabi.lity..Theory. York. 1968. Harcourt. Brace and World. New