Limit distributions for products of sums

Transcription

1 Statistics & Probability Letters 62 (23) 93 Limit distributios for products of sums Yogcheg Qi Departmet of Mathematics ad Statistics, Uiversity of Miesota-Duluth, Campus Ceter 4, 7 Uiversity Drive, Duluth, MN 5582, USA Abstract Let {X; X ; } be a sequece of idepedet ad idetically distributed positive radom variables ad set S = X for. This paper proves that properly ormalized products of the partial sums, ( S =! ) =A, coverges i distributio to some odegeerate distributio whe X is i the domai of attractio of a stable law with idex (; 2]. c 23 Elsevier Sciece B.V. All rights reserved. MSC: 6F5 Keywords: Stable laws; Product of sums; Limit distributio. Itroductio Let {X; X ; } be a sequece of idepedet ad idetically distributed radom variables ad dee the partial sum S = X for. I the past cetury, the partial sum S has bee the most popular topic for study. Such well-kow classic laws as the distributioal laws, stroglaws of large umbers, ad the law of the iterated logarithm all describe the partial sum. I this paper we assume that X. I a recet paper by Rempa la ad Weso lowski (22) it is showed uder the assumptio E(X 2 ) that ( S ) d e 2N ; (.)! where N is a stadard ormal radom variable, = E(X ) ad = = with 2 = Var(X ). Obviously (.) provides a alterative method for the iferece of. The study of (.) was motivated by a study by Arold ad Villaseñor (998) who cosidered the limit distributios for sums of records. Tel.: address: yqi@d.um.edu (Y. Qi) /3/$ - see frot matter c 23 Elsevier Sciece B.V. All rights reserved. doi:.6/s67-752(2)438-8

2 94 Y. Qi / Statistics & Probability Letters 62 (23) 93 Arold ad Villaseñor (998) focused o a special case whe X is a uit expoetial radom variable. I this case, = S d 2N e : (.2) The origial result i Arold ad Villaseñor (998) was stated i a equivalet form with a logarithm take o both sides of (.2): log S log + 2 d N: (.3) This shows that the asymptotic behavior for the products of sums of positive radom variables resembles that for sums of idepedet radom variables uder certai circumstaces. The preset paper focuses o the study of the limit distributios for products of sums of positive radom variables i more geeral settigs. Precisely speakig, we will assume that the positive radom variable X is i the domai of attractio of a stable law with idex (; 2]. The mai result ad its proof will be give i the ext sectio. 2. Mai theorem As i the itroductio, we let {X; X ; } be a sequece of idepedet ad idetically distributed positive radom variables ad set S = X for ad assume X is i the domai of attractio of a stable law with idex (; 2]. Note that E(X ) whe X is the domai of attractio of a stable law with a idex (; 2]. Recall that a sequece of idepedet ad idetically distributed radom variables {X; X ; } is said to be i the domai of attractio of a stable law L if there exist costats ad B R such that S B d L; where L is oe of the stable distributios with idex (; 2]. The followigtheorem is well kow (see e.g., Hall, 98 or Bigham et al., 987). (2.) Theorem 2. (Stability Theorem). The geeral stable law is give, to withi type, by a characteristic fuctio of oe of the followig forms: (i) (t) = exp{ t 2 =2} (ormal case, = 2); (ii) (t) = exp{ t ( i(sg t) ta } ( or 2), 6 6 ; 2 (iii) (t) = exp{ t ( + i(sg t)2= log t } ( =; 6 6 ). It is worth metioigthat i Theorem 2., is the skewess parameter. I our paper, = sice X is a o-egative radom variable.

3 Y. Qi / Statistics & Probability Letters 62 (23) Let F deote the distributio fuctio of X = X. Dee the geeralized iverse of =( F) by { } U(x) = if t: F(t) x : Write S(x)=E[(X ) 2 ]I( X 6 x); for x ; ad deote the geeralized iverse of x 2 =S(x) byv (x) { } t 2 V (x) = if t: S(t) x : Oe ca always take B = i our settig. Throughout this paper we will take = U() if 2, ad = V () if = 2. The from Loeve (977), the limit L i (2.) has a characteristic fuctio as i Theorem 2.. Theorem 2.2. Assume that the o-egative radom variable X is i the domai of attractio of a stable law with idex (; 2] with = E(X ). The costats are deed as above so that the limit L i (2.) has a characteristic fuctio as i Theorem 2.. The ( S! where ( +)= x e x dx. ) =A d e ( (+)) =L ; (2.2) Remark. Whe =2, X is said to be i the domai of attractio of the ormal. A special case is whe E(X 2 ). Sice S(x) 2 =Var(X )asx ;. Moreover, (+)= (3)=2. Therefore, our result coicides with that of Rempa la ad Weso lowski (22). Before we proceed to the proof of the theorem we give the followig lemma. Lemma 2.3. Uder the coditios of Theorem 2.2 log( + ) (X ) d ( ( + )) = L: Proof. We uify the proofs for the cases 2 ad = 2. First let A(x) = U(x) ad D(x) ==( F(x)) if 2, ad A(x) =V (x) ad D(x) =x 2 =S(x) if = 2. Accordigto Loeve (977), whe 2; F(x) is regularly varyig with idex, ad thus =( F) is regularly varyigwith idex ; whe =2, S(x) is slowly varyigad x 2 =S(x) is regularly varyig with idex 2. Therefore, we have that D(x) is regularly varyig with idex (; 2], ad that A(x) isthe geeralized iverse of D(x) ad is regularly varyig with idex = [=2; ). (See, e.g., Bigham et al., 987, Theorem.5.2). Moreover, D(A(x)) A(D(x)) x as x : (2.3)

4 96 Y. Qi / Statistics & Probability Letters 62 (23) 93 Let f deote the characteristic fuctio of X i. The (2.) is equivalet to the followig covergece ( ( )) t f log (t) locally uiformly over t R: A() (See, e.g., Bigham et al., 987, Lemma 8.2.). Sice A(x) is regularly varyig, we have as x ( ( )) t x f log (t) locally uiformly over t R: A(x) The substitutig x by D(s) we have from (2.3) the followigcosequece: as s ( ( t D(s) f log (t) locally uiformly over t R: (2.4) s)) Let f deote the characteristic fuctio of ( ) log( +) ( f (t)= f t = f We will show that for ay t R, log( + ) t =log(( +)=) (X ). The ) : lim f (t)=((t)) (+) : First, ote that =log(( +)=) =log( + ) for all 6 6, ad lim if : (2.5) (2.6) The, from (2.4) we get for ay xed t ( ( )) t D( =log(( +)=)) f log (t) =log(( +)=) uiformly for 6 6, or equivaletly ( ) t + o() f =+ log (t) =log(( +)=) D( =log(( +)=)) where o() is a term that teds to uiformly over as teds to iity. Without ay further otice, we use o() i the sequel to deote some term with such a property. Sice + x =e x+o(x) as x, we get ( f t =log(( +)=) ) { = exp + o() log (t) D( =log(( +)=)) ad hece we coclude for ay xed t R f (t) = exp ( + o()) log (t) D( =log(( +)=)) : }

5 Y. Qi / Statistics & Probability Letters 62 (23) Therefore, i order to show (2.5), it suces to prove ( +): (2.7) D( =log(( +)=)) Sice D(x) is regularly varyig with idex, it ca be writte as D(x)=x L(x), where L(x) is slowly varyig. The (log(( +)=)) L( ) = D( =log(( +)=)) D( ) L( =log(( +)=)) : From (2.3), D( )=D(A()). Thus, D( =log(( +)=)) = ( + o()) (log(( +)=)) L( ) L( =log(( +)=)) : For ay xed small, we will break the right-had side of the above equatio ito three sums: (log(( +)=)) L( ) L( =log(( +)=)) = 66( ) ( ) 6 (log(( +)=)) (log(( +)=)) (log(( +)=)) L( ) L( =log(( +)=)) L( ) L( =log(( +)=)) L( ) L( =log(( +)=)) =I + II + III: By properties of slow variatio, as s, L(sx)=L(s) uiformly over x C, where C is ay compact iterval i (; ). Therefore, (log(( +)=)) I = ( + o()) ( log x) dx: 66( ) To estimate II ad III, we eed the Potter s boud for slowly varyigfuctio L(x): L(x)=L(y) 6 max(x=y; y=x) whe x ad y are sucietly large (Bigham et al., 987, Theorem.5.6). Hece, we get II 6 (log(( +)=)) + ( log x) + dx; ad III (log(( +)=)) ( log x) dx:

6 98 Y. Qi / Statistics & Probability Letters 62 (23) 93 So we get for all small lim sup (log(( +)=)) 6 2 ( log x) + dx +2 L( ) L( =log(( +)=)) ( log x) dx; ( log x) dx which ted to as. Therefore, (log(( +)=)) L( ) L( =log(( +)=)) ( log x) dx = x e x dx: That proves (2.7) ad (2.5). Fially, we eed to show that (t) (+) is the characteristic fuctio of ( +) = L. For (; 2], it is obvious from the expressio of that (t) (+) = ( ( +) = t). That completes the proof of the lemma. Proof of Theorem 2.2. It is easily see that for some costat c (; ) log( + ) ( 2 ) c; which, coupled with (2.6), yields log( + ) =A : Thus, it suces to show log( + )S d =A e ( (+)) =L : By stroglaw of large umbers, with probability oe =: log( + )S ( =log + ) ( S + log + ) (2.8) as. Notice that E X r for all r. For our purpose, we x r (2=( +);). We have by Marcikiewicz-Zygmud s strog law of large umbers (see, eg., Chow ad Teicher, 988, p. 25), S =o( =r ) almost surely. The ((S )=) 2 =o( 2=r 2 ) almost surely. Therefore, [ ( log + )] 2 ( ) S 2 =o( 2=r ) ad [ ( log + ) 2 ] = O(log ):

7 Y. Qi / Statistics & Probability Letters 62 (23) We coclude that with probability oe [ ( log + )] 2 ( ) S 2 +2 [ ( log + ) 2 ] =o( 2=r ): Sice 2=r = ad A(x) is regularly varyig with idex =, ad thus lim x x2=r =A(x)= from property of regular variatio. So we get 2 (2.9) with probability oe. I view of (2.8), log( + )S =+ =e +O(2 ) ; ad thus from (2.9) log( + )S { =A = exp ( )} { +O 2 = exp } + o() : The remaiigtask is to show that d ( ( + )) = L: (2.) As a matter of fact, = = ( log + ) (S )+ ( ( log + ) ) log( + ( ) ) log (X )+O log( + ) = (X ) + o(): (2.) is proved by applyiglemma 2.3. That completes the proof. 3. Ope problem Oe would be iterested i asymptotic behavior of the product of sums whe X is i the domai of attractio of a stable law whe the idex (; ]. Ufortuately we are uable to prove whether (2.2) holds for (; ]. We guess (2.2) could be true whe = but E(X ). For case (; ) (this implies E(X ) = ) or = but E(X ) =, oe has to d some appropriate ormalizatio costats for the product of the sums. This is a usolved problem as well.

8 Y. Qi / Statistics & Probability Letters 62 (23) 93 Ackowledgemets The author is grateful to the referee, whose valuable commets led to improvemet of the layout ad the readability of the paper. The author would also like to thak Professor B.C. Arold for sediga copy of his paper. Refereces Arold, B.C., Villaseñor, J.A., 998. The asymptotic distributio of sums of records. Extremes (3), Bigham, N.H., Goldie, C.M., Teugels, J.L., 987. Regular Variatio. Cambridge Uiversity, New York. Chow, Y.S., Teicher, H., 988. Probability Theory: Idepedece, Iterchageability, Martigales, 2d Editio. Spriger, New York. Hall, P., 98. A comedy of error: the caoical form for a stable characteristic fuctio. Bull. Lodo Math. Soc. 3, Loeve, M., 977. Probability Theory, I, 4th Editio. Spriger, New York. Rempa la, G., Weso lowski, J., 22. Asymptotics for products of sums ad U-statistics. Elect. Comm. Probab. 2,