Extend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables

Transcription

1 ppled Mathematcal Sceces, Vol 4, 00, o 3, xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom Bahram Sadeghpour Departmet of Statstcs, Faculty of Basc Scece Babolsar-Ira, O Box sadeghpour@umzacr Gholam Hosse Yar ad Farhad Hosse Zadeh Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira Yar@ustacr - fo@hossezadehr bstract I ths paper we dscus o the Borel-Catell lemma whe the codto of depedet evets s replaced by egatve quadrat depedet codto Keywords: Borel-catell Lemma, Cauchy-schwarz Iequalty, Negatvely Quadrat Depedet(NQD) Itroducto I the probablty theory, we ofte wsh to uderstad the relato betwee evets the same probablty space The frst ad secod Borel-Catell Lemma ad Fatou's Lemma are the mportat coceptos of probablty theory They have umerous applcatos probablty theory We are ofte terested the lmts superor ad lmts feror of a sequece of evets o the same

2 638 Der, B Sadeghpour, Gh H Yar ad F H Zadeh probablty space ( Ω Α, Ρ), Here we prove the above lemmas whe the essetal codto o evets (e depedet) s replaced by NQD I the secto two we recall some prelmares deftos ad the classcal form of Borel-catell lemma the ma result of our studes relmares Defto For a sequece of subsets, defe the lmt superor lm lm sup { ω : ω for ftly may 's} { ω : ω for ftly ofte} {, o} IU k k Smlarty the lmt feror s gve by lm Remark Note that f lm f { ω : ω for all but ftely may 's} { ω : ω for evetually} {, ev } UI k k Α, are measurable, the so are lm sup ad lm f By Demorga,s law, we have that {, ev } {, o} c, that s, ω for all large eough f ad oly f ω for ftely may ' s lso, f ω evetually, the certaly ω ftely ofte, that s lm f lmsup The lmsup ad lmf of the dcator fuctos o the sets satsfy the followg relatos [,3 lm sup lm f I I ( ω) I ( ω) lm sup ( ω) I ( ω) lm f Lemma ([) ( Borel-Catell Lemma I ) Suppose ( ) < The, Α ad

3 xted the Borel-Catell lemma 639 (, o) 0 Lemma ([) (Borel-Catell Lemma II) Suppose depedet d ( ) The, (, o) Α are mutually Lemma 3 ([) (Fatou's Lemma) Let { } ( Ω, Α, Ρ) The, ( lm sup ) lm ( ) ad sup be evets a probablty space ( ) ( lm ) lm f f Defto ([7) sequece of radom varables { } parwse egatve quadrat depedet (NQD) f, ( X x, X y) ( X x) ( X y) X ; s sad to be For all x, y R ad for all,, For fdg more detals ad results you ca revew [, [4, [5, [6 papers I order to prove the ma theorem, we shall state the followg lemma for later refereces I more cases that we ca studdg about t that to decreases codto depedece low 0- Borel ad ga same result we exteto Theorem, secod Borel-catell Lemma 3 Ma result Lemma 4 Let 0 ad [ Y < Y ad [ Y > a [ Y ( [ Y - a ) a Y roof By apply the cauchy-schwarz equalty [3, oe gets or equvaletly, Sce, The ( Y I) { } / { ( )} / Y a Y I Y a > >, { ( Y I )} [ Y > a [ Y [ Y >a

4 640 Der, B Sadeghpour, Gh H Yar ad F H Zadeh It follows that, Y I [ > a 0 Y f f Y a Y Y I, [ Y> a Y - a Y > a ( ) a, whch, mples Y I[ Y> a Y - ad therefore, [ Y ( - a) [ Y [ Y > a Theorem Let { } ( Ω, Α, Ρ) ad set lmsup If ( ) the, (a) α be a sequece of evets a probablty space lm sup ( [ ) I (b) If the evets are parwse (NQD) the ad, [ α > 0 roof (a) Let Y I be the dcator fucto of the evet I addto, set X X Y k k ad Z It s clear that [ Z, ad, X [ Z Z Z [ Y Y [ X ( [ Y ) Y Y [ ( [ ) Sce Z as defed above s a radom varable that satsfes [ Z, [ Z <, for all,, the for ay η <, we ca use lemma 4 ad wrte, ( [ Z η ) ( η) Z > η, Z Z to establsh the frst result s the followg, ( ) [ sup [ lmsup Z > η lm

5 xted the Borel-Catell lemma 64 To uderstad why ths s so essetally, we prove that Ths gves, c { ω lm sup Z ( ω) > η} c : { ω lm ( ω) > η} : sup Z ssume that ω Ths meas that lm ( ω), s a fte umber ed, thus, ( ω) 0 lm Z, sce by assumpto, X [ X [ as, Ths s the same as sayg that for ay η > 0, ( ω) η o must satsfy the requremet Z ( ω ) > η o or, say Now usg by Lemma 4, we ca wrte, { ω : lm sup Z ( ω) η} ω > Z Hece, fω, t [ lm sup Z > η lm sup [ Z > η lm sup ( η ) α η Usg the secod assumpto of the theorem Sce Is a arbtrary umber (0, ), t follows that α as we were supposed to show (b) If we troduce the extra assumpto that the evets { } are parewse NQD, the ( [ ) I [ + Z [ + [, [ + [, [, [ + [ [, [ ( [ ) [

6 64 Der, B Sadeghpour, Gh H Yar ad F H Zadeh Now sce, [ ( [ ) [ By assumpto, ( ) [ [ [ 0 as, we ca use the statemet ust prove to wrte lmsup ( [ ) I [ Hece Refereces [ Blgsly, robablty ad measure, thrd dto, New York, (995) [ HW Block, TH Savts ad MF Shaked, Some cocepts of egatve depedece probab, 0 (98) [3 K L Chug, coure probeblty theory Harcourt, New York, (969) [4 K J Dev ad F roscha, egatve asso cato of radom varables, wth applcatos Satstll (983) [5 D Dubhash ad D Raa, Balls ad Bs, study egatve depedece Mauscrpt (996) [6 N brahm, ad M Ghosh Multvarate egatve depedece Comm Statst Theory methods 0 (98) [7 Lehma, some cocepts of depedece Math Satst 37 (996) Receved: March, 009