Expectation of the Ratio of a Sum of Squares to the Square of the Sum : Exact and Asymptotic results

Transcription

1 Expectatio of the Ratio of a Sum of Square to the Square of the Sum : Exact ad Aymptotic reult A. Fuch, A. Joffe, ad J. Teugel 63 October 999 Uiverité de Strabourg Uiverité de Motréal Katholieke Uiveriteit Leuve

2

3 Abtract Let X i, i =... be a equece of poitive i. i. d. radom variable. Defie R := E X + X X (X + X X ). Let φ() = Ee X. We give a explicit repreetatio of R i term of φ ad with the help of the Karamata theory of fuctio of regular variatio we tudy the aymptotic behaviour of R for large. Réumé Soiet X i, i =... ue uite de variable aléatoire poitive équiditribuée, R := E X + X X (X + X X ), et φ() = Ee X. Nou doo ue expreio explicite de R à l aide de φ et grâce à la théorie de Karamata de foctio à variatio régulière ou décrivo le comportemet aymptotique de R pour le grade valeur de.

4

5 . Itroductio. Let X, X... X... be a equece of idetically ditributed idepedet o-egative radom variable atifyig P (X = ) <. Let R := E X + X X (X + X X ) with the covetio =. The aymptotic behaviour of R ha bee ivetigated i [CH] ad [MO]. I ectio a explicit formula i etablihed repreetig R i term of the Laplace traform ϕ of the X i : ϕ() := Ee X = e x df (x),, where F deote the ditributio fuctio of X. I ectio 3 the power of the repreetatio formula i illutrated by givig elemetary proof of kow reult. The repreetatio formula allow u both to provide ew ad impler proof of kow reult ad to exted them. For itace theorem (5.4) give ew characteriatio of law which are i the domai of attractio of a table law of parameter α (, ). Our method relie o the theory of fuctio of regular variatio for which [BGT] i our mai referece. I the appedix we ummarize the reult we ue from [BGT] ; give thee, thi article i completely elf cotaied. I ectio 4 ad 5 everal theorem are tated cocerig the aymptotic behaviour of R. I ectio 4 thee are of a geeral ature, ectio 5 give a complete decriptio of the behaviour of R whe the X i are the domai of attractio of a table law. Theorem 4. ad 4. are light improvemet of theorem of [MO], theorem 5. ad 5. are the theorem. of [MO], theorem 5.5 i theorem of [CH] ad theorem 4 of [MO], the firt part of theorem 5.4 i the firt part of theorem 4 of [CH],whoe proof i omitted, ad the ecod part of theorem 5.4 ad theorem 5.6 are ew. Mot of thee reult were tated without proof i [FJ] though there i a mall icogruity i their theorem.. Repreetatio formulae. Let X ad Y be o-egative radom variable with Laplace traform ϕ X (), ϕ Y () ad joit Laplace traform ϕ X,Y (, t). The formula y α = e y α d, α >, (α) ad Fubii theorem yield E Y α = α ϕ Y ()d, (α) ad for k ad k o-egative iteger : E Xk Y k (X + Y ) α = ( )k +k α k +k (α) k k t ϕ X,Y (, t) =t d. I the pecial cae whe X,..., X are idepedet radom variable thi give X k E (X X ) α = ( )k α ϕ (k) X (α) () ϕ Xi ()d, α >, ad fially, whe they have the ame ditributio, thi yield by ymmetry the repreetatio formulae : E Xk X k (X X ) α = ( )k α ϕ (k) X (α) ()ϕ X ()d R = E X X (X X ) = Thi lat formula i relevat for our problem. ϕ X ()ϕ X ()d. (.)

6 3. Some elemetary reult. Let u retur to the iitial ituatio i which the X i are i. i. d. o egative r. v.. There commo Laplace traform i deoted by ϕ ad the tartig poit of mot of the proof i the above repreetatio formula (.). Proof of the followig lemma are eay ad are left to the reader. Let µ := EX if EX <, d() := ϕ() Lemma 3.. If E(X ) <, the lim R = E(X ) [E(X )]. PROOF. It uffice, from lemma 3.5, to etablih the reult for the expreio : ɛ A = ϕ ()ϕ ()d = ɛ tϕ ( t )ϕ ( t )dt. With the help of Lemma 3.4, the itegrad i ee to coverge to te(x )e µt. Uig Lemma 3.3 ad the mootoocity of ϕ ad ϕ, the reult follow by Lebegue domiated covergece theorem. Theorem. If EX <, the lim R =. PROOF. It uffice,from lemma 3.5, to etablih the reult for A. Lemma 3. implie that the itegrad coverge to. The reult follow by Lebegue domiated covergece theorem ice the aumptio implie that ϕ () i itegrable ad with the help of Lemma 3.3, the followig upper boud i eaily obtaied, for ϕ () ϕ () ϕ(ɛ) ϕ ()e ϕ () ϕ(ɛ)ϕ (ɛ) e ϕ(ɛ)ϕ (ɛ). 4. Aymptotic behaviour of R : geeral reult. The ext reult i a lightly more precie verio of Theorem of [MO] : Theorem. lim up R x[ F (x)]dx if < EX µ if µ := EX. PROOF. It uffice from lemma 3.5 to etablih the reult for A. That ϕ i decreaig implie hece from lemma 3.3 A ϕ(ɛ) ɛ ϕ ()e d(ɛ) d A ϕ(ɛ) ϕ(ɛ) ɛ ϕ ()ϕ ()d, x df (x) e (x+d(ɛ)) d = ϕ(ɛ) (D + E ),

7 where a) E ( ) x D = df (x), E = x + d(ɛ) df (x) = [ F ()], ( ) x df (x). x + d(ɛ) b) D [d(ɛ)] x df (x) [d(ɛ)] x d[ F (x)] = d(ɛ) x[ F (x)]dx [ F ()] d, (ɛ) hece D + E d x[ F (x)]dx + ( (ɛ) d )[ F ()]. (ɛ) If µ the d(ɛ) µ for every ɛ >, ad we get D + E d x[ F (x)]dx, (ɛ) which give the reult ice, a ɛ goe to, d(ɛ) goe to µ ad ϕ(ɛ) goe to. If EX >, there exit ɛ > uch that d(ɛ) >, hece if we oberve that [ F ()] x[ F (x)]dx, we get which give the reult. D + E x[ F (x)]dx, Theorem. If µ = EX <, the lim if ( + µ) R x[ F (x)]dx PROOF. A i the proof of Theorem 3. it uffice to etablih the iequality for the expreio ɛ ɛ A := ϕ ()ϕ ()d ϕ ()ϕ ()d. From lemma 3.4, for ay δ > oe ca chooe ɛ > uch that ϕ() e (µ+δ) for < ɛ. Hece ɛ A = ( ( ɛ ) ϕ ()e (µ+δ) d = x df (x) e [x+(µ+δ)] d ) df ɛ[x+(µ+δ)] ( (x) x x + (µ + δ) 3 ye y dy x x + (µ + δ) ) df (x);

8 breakig the itegral from to ito the part D ad from to ito the part E oe obtai : ( x ) df a)e := (x) x + (µ + δ) ( (µ + δ) ) df (x) = [ F ()]; + (µ + δ) [ + µ + δ] ( x ) df b) D := (x) x + (µ + δ) [ + (µ + δ)] x df (x); itegratio by part yield : D ( + µ + δ) x[ F (x)]dx Fially, A i ultimately bouded below by D + E : D + E ( + µ + δ) [ F ()]. ( + µ + δ) x[ F (x)]dx, from which the reult follow ice δ > ca be choe a mall a we wat. 5. Aymptotic behaviour of R : pecial reult.. Referece to Th. A. i refer to the appedix (ectio 6). Theorem. If X belog to the domai of attractio of a table law of parameter α, α < ad µ = EX <, the : lim R [ F ()] = ( α) ( + α) µ α PROOF. It uffice,from lemma 3.5, to etablih the reult for the expreio : B = The chage of variable = t yield B = Hece with the otatio of Th. A.6 : B = α( α) = h (t)dt, F () F () l() α [ F ()] ɛ ɛ ɛ t α l( t ) l() ϕ ()ϕ ()d. tϕ ( t )ϕ ( t )dt. which defie h. From Th. A.6 we have, for every t > ϕ ( t ) α( α)( t ) α l( t ) } {{ } ϕ ( t ) dt e µt lim h (t) = α( α)t α e µt. Uig Potter theorem (Th A.), Th. A.6 ad lemma 3.3, the paage to the limit i jutified by Lebegue domiated covergece theorem. Fially h (t)dt α( α) t α e µt dt = ( α)( + α) µ α. 4

9 Theorem. If X belog to the domai of attractio of the ormal law, the lim R x[ F (x)]dx = µ PROOF. It uffice,from lemma 5, to etablih the reult for the expreio : C = ɛ x[ F (x)]dx ϕ ()ϕ ()d. The chage of variable = t give C = Uig the otatio of Th. A.5 : C = ɛ x[ F (x)]dx l() ɛ x[ F (x)]dx t l( t ) l() ϕ ( t ) l( t ) tϕ ( t )ϕ ( t )dt. ϕ ( t ) dt = e µt h (t)dt, which defie h. Itegratig by part ad uig part a) of Theorem A.5. give : x[ F (x)]dx = [ F ()] + l(). Therefore x[ F (x)]dx = [ F ()] + l() l(), which ted to a, by Theorem A.5. A i the proof of the previou theorem, the reult follow from Lebegue domiated covergece theorem : h (t)dt te µt dt = µ. Theorem 5. ad 5. are liked. For if the coditio of 5. are atified, the { ( F ()) x( F (x))dx = Hece we obtai the followig corollary : y F (y) } F () dy α. Corollary. If X belog to the domai of attractio of a table law of parameter α, α ad µ = EX < the, lim R (3 α) ( + α) = x[ F (x)]dx µ α The followig lemma will be ued i the proof of ext theorem. 5

10 Lemma. If X belog to the domai of attractio of a table law of parameter α, < α, or if X i relatively table the i) There exit a equece uch that ϕ ( ) e α a, with α = if X i relatively table. ii) There exit δ with < δ < ad ɛ > uch that : ( ) t ϕ e Atδ, t ɛ, > ɛ PROOF. Coider firt the cae < α <. The are choe a i the commet which follow Th A.6 ; by that theorem oe ha : F (x) l(x) ( α) x,with l lowly varyig, which yield by α Th. A.4. b) ϕ() = α l( ϕ() ). The d() = = α l( ) ad lemma 3.3 yield : Therefore : Now by (6.3 ), l() a α ϕ() e d() e α l( ). ϕ( ) e α l( ) a α, ϕ ( ) e α l( ) l(a) l(a) a α.. The reult follow by applyig Th. A.. (Potter theorem) to the ratio l( ) l( ), with ɛ = X. The cae α = i treated i the ame way by ivokig Th. A.8 itead of Th. A.6. ad A.4. Oberve that l play the role of l. Fially whe X i relatively table, the reult follow from Th. A.9. Theorem. If X α, < α <, the belog to the domai of attractio of a table law of parameter lim R = α. ( ) Coverely if ( ) hold the X belog to the domai of attractio of a table law of parameter α, < α <. PROOF. A before, to tudy the aymptotic behaviour of R it uffice,from lemma 3.5, to etablih the reult for A. From the remark followig Th. A.6, the aumptio implie the exitece of a equece, > uch that ϕ ( ) e α a. After the chage of variable = t, we get a ɛ A = a tϕ ( t )ϕ ( t )dt. Equatio (6.4) ad (6.5) of Theorem A.7 lead to : A = a a ɛ = a α l( ) = tα( α)( t ) α l( h (t), a ɛ α( α) t ) t α l( t ) l( ) ϕ ( t ) α( α)( t ) α l( t )ϕ 6 ϕ ( t ) ( t )dt α( α)( t ) α l( t ) ϕ ( t ) dt a }{{ } e tα

11 which defie h. Formula (6.3) ad Th. A.7 yield : lim h (t) = α( α)t α e tα. Let ɛ be choe a i lemma 5.3 ; with the help of that lemma, the reult follow from Lebegue domiated covergece theorem. To prove the covere, let u aume that R coverge to c, < c < a goe to. For ay iteger m we have R m+ = (m + ) ϕ ()ϕ m ()d = (m + ) ϕ m ()d(), where () = tϕ (t)dt. The chage of variable ϕ() = u, ( = ϕ ( u )), yield R m+ = (m + ) ( u )m d (ϕ ( u ) ). Defie µ (u) = (ϕ ( u ) ), which are meaure o R + of ma. Sice µ (u) u weak compacte implie the exitece of a ubequece i uch that µ i (u) µ(u) weakly. R m+ = (m + ) takig the limit alog the ubequece i yield From c = m ( u )m dµ (u), e um dµ(u). e um dµ(u) = c m, m =...., ad the chage of variable x = e u it follow by Weiertra theorem that µ(u) = cu, idepedetly of the ubequece. Hece µ (u) cu weakly or µ (u) = (ϕ ( u ) ) cuor () c[ ϕ()]a. Thu (t)dt tϕ c or tϕ (t)dt + c ϕ (t)dt = o( ϕ()). ϕ() Call the left had ide of thi lat equatio V () : V () := tϕ (t)dt + c ϕ (t)dt = t c [t c ϕ (t)] dt = ϕ V () () ( c)[ϕ() ], whece ϕ() = ϕ () + c = o(), a. ϕ() Thi yield ϕ () ϕ() c a. Put δ() := c + ϕ () ϕ(), t to obtai δ() d = ( c) log t + C + log[ ϕ(t)] or ϕ(t) = t c+ l( t ) where l(u) = exp{ C + u ɛ(x) x dx} ad ɛ(x) = δ( x ). From the repreetatio theorem (theorem A.), it follow that l i a lowly varyig fuctio ad the theorem A.5. ad A.7. imply that F i attracted to a table law of parameter c. 7

12 Theorem. The followig three tatemet are equivalet : () F i relatively table, () x [ F (y)]dy i a lowly varyig fuctio, (3) lim R =. PROOF. () ad () are equivalet by Th. A.9. Aume () ; a uual it uffice to etablih the reult for A. Defie l(x) := x ( F (y)) dy. After a itegratio by part ad a chage of variable, oe obtai : ϕ () = e x x d( F ) = l( y )e y ( 4y + y )dy. Replacig ϕ () by thi value i the expreio of A ad uig Fubii theorem, oe obtai : A = After the chage of variable A = = [ e y ( 4y + y ) e y ( 4y + y ) t where l() = a ɛ [ ɛ ϕ ()l( y )d ]dy. with ϕ ( t ) e t, ϕ ( t )l( y t [ e y ( 4y + y ) ] )dt dy l( ) Let ɛ be choe a i lemma 5.3 ; that lemma ad the relatio e y ( 4y + y )dy = a ɛ ϕ ( t ) a }{{ } e t l( y t ) l( ) the implie that A a by Lebegue domiated covergece theorem. To prove the covere proceed a i theorem 5.3 with c = to obtai ϕ(t) = tl( t ) with l a lowly varyig fuctio ; it follow from Theorem A.4. ii) that x ( F (x)) dx i a lowly varyig fuctio. The lat cae to be coidered i the oe where X belog to the domai of attractio of a table law of parameter α, with α = ad EX = ; the from Theorem A.6 ad A.8 oe ha : F (x) l(x) x, x ad moreover oe ca chooe a equece {}, > uch that [ϕ( t )] e t, ad l() where l(x) = x l(t) t dt. 8 dt ] dy.

13 Theorem. If X belog to the domai of attractio of a table law of parameter α, with α = ad EX = the lim l(a ) l( ) R =, where the ymbol l, l, have bee defie above. PROOF. A uual it uffice to etablih the reult for the expreio : A. After the chage of variable = t we get Uig the otatio of Th. A.8 : A = l() l( ) l(a ) a }{{ } A = a ɛ a ɛ l( t ) l( ) tϕ ( t )ϕ ( t )dt. ϕ ( t ) t l( t ) } {{ } ϕ ( t ) a }{{ } e t dt = l(a ) h (t)dt, l(a ) which defie h ad a i the previou theorem, with the help of lemma 5.3 the reult follow from Lebegue domiated covergece theorem. Remark. I the above theorem ice l() l( ) oe ha the rate of covergece of R to, a. We thak J. Kuelb for akig oe of u a quetio which led to the followig remark : Remark. If X belog to the domai of partial attractio of a table law of parameter α, where < α <, the, alog a ubequece r, lim r R = α. Hece if X belog to Doebli uiveral law the et of all limit poit of R i the cloed iterval [, ]. Appedix. Slowly varyig fuctio ad domai of attractio. Let l be a poitive meaurable fuctio atifyig l(λx) l(x) (x ) λ > (6.) the l i aid to be lowly varyig. If thi i the cae oe ca how that the covergece i () i uiform i each compact-λ et i (, ). Theorem. (i) If l i lowly varyig the for ay choe cotat A >, δ > there exit X = X(A, δ) uch that l(y) { l(x) A max ( y x )δ, ( y } x ) δ (x X, y X). (ii) If, further, l i bouded away from ad o every compact ubet of [, ), the for every δ > there exit A = A (δ) > uch that l(y) { l(x) A max ( y x )δ, ( y } x ) δ (x >, y > ). 9

14 Theorem. The fuctio l i lowly varyig if ad oly if it may be writte i the form x ɛ(u) l(x) = c(x) exp{ u du}, where c(x) c (, ), ad ɛ(x) a x. I the equel F (x) will deote a ditributio fuctio whoe upport i [, [, ˆF () = ϕ() = e x df (x),, it Laplace traform, V (x) = x t df (t), x, the trucated variace of F ( ) ad ˆV () = ϕ () = e x dv (x) = e x x df (x),. Theorem. (i) If l i lowly varyig, α < the the followig two tatemet are equivalet : a) V (x) x α l(x), x, b) F (x) α l(x) α x, x. α (ii) If l i lowly varyig, (α = ), the the followig two tatemet are equivalet : a) V (x) l(x), x, b) x [ F (x)] V (x), x. Theorem. (i) If l i lowly varyig α < the the followig two tatemet are equivalet : l(x) a) F (x) ( α) x, x α b) ˆF () α l( ),. (ii) The followig two tatemet are equivalet (α = ) : a) x [ F (x)] l(x), x b) ˆF () l( ),. The followig two theorem o domai of attractio are a adaptatio of Th of [BGT] whe the upport of F i R + : Theorem. the followig four tatemet are equivalet : a) F i attracted to ormal law (α = ), b) V (x) = l(x) where l(x) i lowly varyig, c) x [ F (x)] V (x) a x, d) ˆV () = ϕ () l( ) a. Theorem. the followig four tatemet are equivalet : a) F i attracted to a table law ( < α < ), b) F (x) l(x) x where l(x) i lowly varyig, α c) V (x) α α x α l(x) a x, d) ˆV () = ϕ () α( α) l( α ) a. We recall from [BGT] ectio ( ) that the table law with Laplace traform e α, o < α <, are (to withi cale) the oly oe cocetrated o [, [. Coider their domai of attractio ad oberve that o ceterig i required : F i i their domai of attractio if ad

15 oly if there exit a equece, with > ad Moreover oe ha F (x) ( α) ad the ormig cotat are determied from l( ) a α + uch that ϕ ( ) e α a. l(x), for ome lowly varyig fuctiol, (6.) xα. (6.3) Theorem. Equatio (6.) i equivalet to each of the followig : α V (x) ( α)( α) x α l(x), x. (6.4) ˆV () = ϕ () α( α)( ) α l( ),. (6.5) Whe α = the ituatio i lightly more techical : recall from [BGT] that the cla l of l idex equal to i the cla of fuctio f uch that for λ >, {f(λx) f(x)} lim x = log λ. l(x) If l(x) i lowly varyig, the i lowly varyig ad l(x) l(x) l(x) = x l(t) t dt ax. Theorem. If l i lowly varyig the the followig four tatemet are equivalet : a) F (x) l(x) x, x b) ϕ () l( ), c) ϕ() d) ϕ() l of l idex equal to l[t] t dt = l( ),. Moreover if F i i the domai of attractio of a table law of parameter, oe ca chooe a equece { }, > uch that l(a ) o that [ϕ( t )] e t, PROOF the equivalece of a) ad b) i Th. A.6 for α =. Thi i equivalet to c) by the remark i [BGT] p.335 ad the lat two tatemet are equivalet by Th i [BGT] p.58. Fially the lat tatemet follow from Th of [BGT] pp ice oe ca chooe a equece uch that l(), thi yield the cocluio. Recall that a law F i relatively table if there exit ormig cotat uch that lim ϕ ( ) = e. The followig theorem i a adaptatio of Theorem ad formula (8.8.) of [BGT].

16 Theorem. The followig three tatemet are equivalet : i) F i relatively table, ii) x ( F (y))dy l(x), x where l i lowly varyig, iii) ϕ() l( ), where l i lowly varyig. Moreover if F i relatively table the ormig cotat atify : l( ) REFERENCES [BGT] N. H. BINGHAM, C. M. GOLDIE ad J. L. TEUGELS. Regular Variatio. Cambridge Uiverity Pre, 987. [CH] H. Coh ad P. Hall. O the Limit Behaviour of Weighted Sum of Radom Variable, Z. Warcheilichkeittheorie verw. Gebiete., t. 59, p , 98. [F] W. Feller. A Itroductio to Probability Theory ad it Applicatio. vol. II, d ed. Wiley, N. Y., 97. [FJ] A. FUCHS ad A. JOFFE. Formule exacte de l epérace du rapport de la omme de carré par le carré de la omme, Compte Redu de l Ac. de Sc., t. 35, p , 997. [MO] D. L. Mc LEISH, ad G. L. O BRIEN.. Expectatio of the Ratio of the Sum of Square to the Square of the Sum, A. Probab., t., p. 9 8, 98.