Sums of independent random variables and sums of their squares

Pub . Ma . UAB N° 22 No . 1980 Ac es VII JMHL SUMS OF INDEPENDENT RANDOM VARIABLES AND SUMS OF THEIR SQUARES E a is Giné Seccidde Ma emá iques Uni e si a Au ónoma de Ba celona Le {Xni :i= 1, ._k n , n e YO be a iangula a ay o ow-wise inde- penden andom a iables, Sn= EJXnj he ow sums and Tn= E j x2j he ow sums o squa es . Raiko (1938) p o ed ha S con e gesweakly o a Gaussian law i and only i Tn con e ges in p obabili y o a cons an . Hall (1978) shows ha i Sn con e ges o a Poisson law wi h pa ame e a, hen so does T n . In his no e we gi e he exac ela ion be ween igh ness and con e gen e o {L(Sn)},{L(Tn )} and {min(1,x 2 )E j dL(X ni )} o in ini esimal  a ays ; he- se esul s con ain hose o Raiko and Hall as pa icula cases . The igh - ness ela ions p o ed o be use ul in some wo k wi h M .P .Ma cus on he cen- al limi heo em in C(S) . I acknowledge P o . M .Ma cus o he co espon- dence ha led o his no e (as a byp oduc ) . The no a ion will  be as ollows :  {X nj :  j =1_ .,k n ,  n e i} will  be a iangula a ay o ow-wiseindependen andom a iables ({X nj } o sho ), Sn =E .X n , T =E .X 2 , X  = X I  < T}' Sn,T-EjxnjT' (T>0), J  j  n  J  nj  ni , ni  {IX n i j_ and {Xnj } will deno e independen symme iza ions o {X nj } .{X nj ) is in ini - esimal i lim n max j P{IX ni 1 > E} =0 o all E>0 . IJe e e o Gnedenko and Kolmogo o (1968, Theo em 25 .1) o o A aujo and Giné (1980, Theo em 2 .4 .7) o he gene al cen al limi heo em on he line(CLT) . The ollowina is ou main obse a ion . I elabo a es on exe cise 2 .5 .3 o A aujo and Giné (1980) and i s p oo is inspi edon hei p oo o he con e se CLT . heo em 1 . Le {Xnj } be a iangula a ay o ow-wise independen 's . Then {L(E i xn i )} is igh i and only i he amily measu es is uni o mly bounded and igh . d n (x)= min(l,x 2 )E j dL(X nj )(x) 12 7 P oo . Assume {L(E~X2~)} igh . Then by posi i i y, so a e {L(EiX2iT)}n-1 and {L(Xni )}n,J . The con e se Kolmogo o and Lé y inequali ies gi e whe e c T d is a ini e cons an o each T, d > 0 . The igh ness o {L(Ei(X2jT)-)} he e o e implies sup n E(E j (X2~ T - c _  < hence ness o {L(E .X 2 -E .EX 2 )} by Chebyshe . {L(E .X 2 )} being igh , we con- ,1 niT J ni T ~ niT elude (1)  sup n E~EX2 jT o al] T >O . As is well independen symme ic, hen o al] d> 0(as we can conclude Since which E(E i (X2 jT-EX2iT )) 2 I we he e exis s T > 0 such ha This p o es ha  {E i L(Xnj)lIxlI> T1/2} ce by (1 , ),  so is  {min(1,x 2 /T)E i L(X ni )} some T> 0 hese measu es a e uni o mly o each T >O, in pa icula o - = 1 . Con e sely, assumenow ha ( n } o al l  T,  > 0, 12 8 < CT,d/(1-2P{IEi(X2iT)-j'>d}1 known, Lé y's ineqiali y gi es ha En =1 P{In i I > d} <  -log(1-2P{IE i n i l > 6» obse ed by Felle (1971), page 149) . Hence, ha he e exis B> 0 and xni e IR  such ha supnEjP{IX2J-xni1 >S} < 1/2 . {L(X 2  is igh , he e exis s M> 0 such ha sup_ ;P{X? ;> M}< 1/2, 1J . 1,J  _  11,J 11J implies ha Ix ni ¡ < M+S . So, he e exis s T > 0 such ha sup n E i P{X2 i> T} < 1/2 . i {n i } a e using Fubini apply his o independen copies o Sn , e1N, we conclude ha sup n E j P{X2 i >T } < 1/2 . P{T n > }<  P{E i X2 iT > ,  IXn il < T, J=1, . . .,kn } he igh - is uni o mly bounded and igh , hen- I is i ial o see ha i o bounded and igh , he same is ue is uni o mly bounded and igh . Then +EiP{ I XnJ l > T} <  EjEX2iT/ +EiP{ I Xn .l > T}  . Gi en e > 0 choose T > 0 such ha he las sum is no g ea e han e/2 and hen such ha E j EX 2 / <e/2 . Hence, {L(T n )} is igh .0 Since (2)  E(Xni-EXnj1) 2 1{IXnJI <l}-EX 2 njl = he p e ious heo em oge he wi h heo em 2 .45 in A aujo and Giné(1980~gi e : Co olla y 2 . Le {Xnj } be an in ini esimal a ay such ha (3)  sup n Ej (EX ni1 ) 2< m . =-(1+p{IXn .i1 > l})(EXnJl)2, Then, {L(Sn-ESn,T)} is igh i and only i {L(E j X2 j )} is . Rema k . Condi ion (3) is sa is ied i : (a) {X nj } is symme ic, bu in his case i is no necessa y o assume in- ini esimali y (use A aujo and Giné (1980) . Co . 2 .5 .7), and (b)  o {Z ni= Xn j-EX njT }, any T > 0,  i ei he {L(S n -ESn,T )} o {L(E i (X ni -EX njT ) )} a e igh o some  T>0 . Le us see i o T = 1 1 EZnj1l < l1'IXni1 <1ZnidPi +(1+ [EXn j1l)P{IX n jl > 1-IEXnjll} = ¡EXn jllp{IX n jl> 1} +(1 +¡EX n jll)P{IX n jl > 1-IEXnjll}, and since max j IEX njl l->0 as n- by in ini esimali y, we ob ain ha sup n E j (EZnil ) 2 < c sup n E J P{IX ni 1 > 1/2} and his quan i y is ini e i ei he one o he wo amilies o sums a e igh , by he p e ious heo ems . So we ha e :  - Co olla y 3 .  Le {X ni* } be in ini esimal .  Then  {L(S n -ES n,d )}  is igh i and only i {L(E i (X ni -EXnjd ) 2 )} is igh o some (all) d > 0 . Nex we examine con e gen e ela ions . We will le T(x) = x2 . No e ha i p and a e a- ini e Bo el measu es he equa ion oT =P, unknown, has a uniquesymme ic solu ion and a unique solu ion suppo ed by R + (R - ) . This solu ion will be deno ed = poT and i will  be symme ic o suppo éd by R + depending on he con ex . Theo em 4 . Le {Xnj } be an in ini esimal a ay . (a) Assume {Xnj } sa is iescondi ion (3) and (4)  L(S n -ESn,a )- w N(O,a 2 )*c 6 Poisp 12 9 o some o2> 0, Lé y measu e ) .+ and 6 >0 such ha p{-6,0= 0 . Then, (5)  L(E i X2 i -E i EXñ ja )-> wc6 2 Pois(poT -1 ) In pa icula , i condi ion (3) is eplaced b . y he s onge condi ion (6)  l imn Ej EX2 ja = a < ~, hen (7)  L(EjX2,j)->w6a*cal Pois(poT-1) (b) Con e sely i he X ni a e non-nega i e (symme ic) and (8)  lim ayOlim nE i (EXnja ) 2 = 0, hen, he ac ha (9)  L(E j X2 )_w6a*ca2 Poisp o some Lé y measu e p and a such ha p{-6 2 ,6 2 }= 0, implies ( 10 ) .  L(S n -ES n,a )- w N(O,o 2 )*c 6 Pois(poT), 9 whe e o~ =a- n  xdp(x) .  (poT is symme ic i he Xn a e symme ic and wi h ,, suppo in h + i he X n , a e non-nega i e) . Also, E~EX2~ a ->a . (b')  I in  (9)  ji= 0,  hen  (b)  is ue wi hou he a iables X ni  being non- nega i e o symme ic . P oo .  (a)  By he CLT  ,  Ei L(X nj ) I{Ixi >8} ->wpI{Ix1 >61  i w{-M}= 0 . Then i p{-6 1 / 2 ,6 112 }= 0, E .L(X 2 )1  poT -1 ~  On he o he J  ni {~x~ >6}- w  {ix1 >0* hand, because by condi ion (3) and he CLT  , sup nz j EX2 < - (see (2)) . Hence (5) ollows om he CLT .  nil (b) Assume now ha (8) and (9) hold . Then (3) holds ° and he e o eCo olla y 3 gi es ha {L(S n -ESn,a )} is igh . Bu ob iously al] he subsequen ial limi s ha e he same Lé y measu e poT,  hence E i L(X nj)j { , x , >6}, wpoTj{jxj>6} 13 0 lim64-0limnEj E[XniI{X2J <6} -(EX ni I {X2 i <6})2l 1¡m ay 01¡m n E~(¿ - E(Xn J ) 6 )E(Xn J ) 6 =0 i poT{-8,8}= 0 . Now, he CLT  and (9) gi e lim nz j EX2 =a, and he e o e, condi ion (8) implies limTyO {lim}nEj(EXnjT)2)= limTy,0 {lim}nEjEXniT 2 = 1 im T+0,p{T 2 } = 0l im n Ei EXn jT = 1 im TyO,u{- 2 } = 0 [a- 1? xdp(x)] l 2 =a- 0 xdu(x) . . So, (10) ollows by he CLT .  (b') also ollows om he CLT, Gaussian con e gence, and om (ii) wi h p =0 . Rema ks 41) Condi ion (8) is sa is ied in he symme ic case and also o Znj = X nj -EX nj1 in gene al ( om ema k (b) a e Co olla y 3 we ob ain ha i {L(E j Xn j )} o {L(E j 7ñ j )} a e shi igh , hen limn E j EIZnjd l 2 < <  lim nz j (P{1 Xnj 1 > 6})` < lim n maxj P{lX nj1 >8} , Ej(P{1X n j1 > 8}= 0) . (2) Le us inally ema k ha i bo h {L(S n )} and {L(E i Xni )} con e ge, hen he p- h momen o IS n 1 con e ges i and only i he (p/2)- h momen o E i Xn i does :bo h condi ions a e equi alen o 1im ~sapnziE1XniiPI{IXnj 1 > }=0 (de Acos a and Giné (1978)) . Wi h his ema k, Theo em 5 con ains he esul in Hall (1978) as a pa icula case ( he cases p = 0 and p = x81) . (3) I is also clea om he o egoing ha he powe 2 is basic only i lim nEj EX2 j 0 and he Lé y measu e u (o poT -1 ) gi es posi i e mass o in e als a bi a ily nea o ze o . 0 he wise he p e ious esul s hold o EjlX n j1  ,  o any p > 0 (as obse ed by Hall  (1978)  in he pa icula case p=x8 1 ) . 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