Strong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA
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1 Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics CET BUT BBSR ODISHA INDIA Abstact: 99 atheatics subject classificatio (Ae ath Soc): 6 B 99 Keywods ad phases: Idepedet idetically distibuted ado vaiables ado algebaic polyoial ado algebaic equatio eal oots I INTRODUCTION Let N be the ube of eal oots of the algebaic equatio f () k ξ k whee ae idepedet ado vaiables assuig eal values oly Seveal authos have estiated bouds fo N whe the ado vaiables satisfy diffeet distibutio laws Littlewood ad Offod [] ade the fist attept i this diectio They cosideed the cases whe the ae oally distibuted o uifoly distibuted i (- +) o assue oly the values + ad with equal pobability They obtaied i each case that N μlog logloglog A log Saal [] has cosideed the geeal case whe the have idetical distibutio with eceptio zeo vaiace ad thid absolute oet fiite ad o-zeo He has show that N s log outside a eceptioal set whose easue teds to zeo as teds to ifiity whee s teds to zeo but s log teds to ifiity Saal ad isha [4] have cosideed the case the have a coo chaacteistics fuctio ep C t whee C is a positive costat ad They have show that N μ log loglog outside a eceptioal set easue at ost ' μ (log log )(log ) μ loglog log if if I all the above cases the eceptioal set depeds upo Evas [] was the fist to obtai stog esult fo these bouds I such case the eceptioal set is idepedet of the degee of the polyoial We use the te stog esult i the followig sese: All the above esults ae of the fo N μ as i teds to ifiity wheeas the theoe of Evas is of the fo sup N μ as teds to ifiity Evas [] has show i case of oally distibuted coefficiets that thee eists a itege such that fo N μ log loglog μloglog ecept fo a set of easue at ost log Saal ad isha [5] have show i the case of C t chaacteistic fuctio ep that fo N μ log loglog outside a eceptioal set of easue at ost whee μ log log log log I [7] they have cosideed the stog esult i the geeal case Assuig that the ado vaiables (ot ecessaily idetically distibuted) have eceptio zeo 6 Iteatioal Joual of Reseach i Egieeig ad aageet Techology ISSN: ol Issue July 5
2 Dipty Rai Dhal et al vaiace ad thid absolute oet o-zeo fiite they have show that fo N (μlog)/log (K outside a set of easue at ost μ log log K log log t /t )log K li povided t is fiite ad log t = a (log) whee K = σ t aσ ad a τ σ τ beig the vaiace ad thid absolute ξ oet espectively of log Ou object is to ipove the stog esult fo lowe C t boud i case of chaacteistic fuctio ep We have show that fo N log Outside a eceptioal set of easue at ost μ log whee but log The esult of Evas [] is a special case of ous ad is (loglog ) obtaied by takig = ad i ou theoe The esult of Saal ad isha [5] is also a special case of ou theoe O the othe had ou eceptioal set is salle All authos who have estiated bouds fo N have used oe kid of basic techique oigially used by Littlewood ad Offod [] We shall deote μ fo positive costats which ay have diffeet values i diffeet occueces We suppose always that is lage so that ay iequalities tue whe is lage ay be take as satisfied Thoughout the pape [] will deote the geatest itege ot eceedig It ay be oted that although Evas [] is a special case of ous a uch bette estiate fo the lowe boud with salle eceptioal set ca be deived fo ou theoe Fo eaple if we take = <p< the fo p (loglog ) whee Stog Result fo Level Cossigs of Rado olyoials N log (log log ) p outside a eceptioal set of easue at ost μ(log log ) log p II THEORE Let f(w) be a polyoial of degee whose coefficiets ae idepedet ado vaiables with a C t coo chaacteistics fuctio ep whee = ad C is a positive costat Take {e }to be ay sequece tedig to zeo such that e log teds to ifiity as teds to ifiity The thee eists a itege such that fo each the ube of eal oots of ost of the equatios f()= is at least e log ecept fo a set of μ log easue at ost Lea If a ado vaiable ζ has chaacteistic fuctio C t ep the fo evey e C ξ This lea is due to Saal ad isha [4] ROOF OF THE THEORE Take costat A ad B such that <B< ad A Choose such that ad as teds to ifiity Let () So μ / log μ We defie (X) log Let k be the itege deteied by υ(8k 7) () 8k7 both ted to ifiity Ae B υ(8k ) 8k 7 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5
3 Dipty Rai Dhal et al The fist iequality gives k iequality gives llog μk log log 8k log (8k ) (8k ) (8k ) log So Thus μ k μ log log log The secod (8k ) log(8k ) (8k ) log μ k μ log log () By the coditio iposed o it follows that k teds to ifiity as teds to ifiity We have f () U R at the poits fo X υ(4 ) 4 / (4) = k / k / k w hee U ξxr ξx the ide v agig fo υ(4 ) υ(4 ) υ(4 f ( ) i ad υ(4 ) 4 to fo to to i We also have ) UR f ( ) U R (5) fo Obviously U ad U + ae idepedet ado vaiables Agai it follows fo () that k+< fo lage Also the aiu ide i U + fo =k is υ(8k 7) (5) Let 8k7 which by () is cosistet with / The Stog Result fo Level Cossigs of Rado olyoials φ(4 ) 4 φ(4 ) υ(4) φ(4 ) φ(4 ) φ(4 (e / A) ) φ(4 ) 4 φ(4 ) φ(4 ) 4 (B / A)e (6) whe is lage Now we estiate U U U U U ( U U ) ( U ξ Sice the chaacteistic fuctio of is ep C t the chaacteistic fuctio of U is theefoe C t ep C t ep 4 whee the ide ages fo 8 υ(8 ) to υ(8 ) i 4 Thee foe the chaacteistic fuctio of U / is ep C t which is siilaly also the chaacteistic fuctio U / Thus the chaacteistic fuctio is depedet o Let F() be the coo distibutio fuctio Hece U ) ( U / ( U / ) F() Thus = F() F( ) F( ) F() F( ) F() δ(say Obviously δ ) We shall eed the followig leas Lea ost lage ξ / CAe ep - (4 ) B( ) ecept fo a set of easue at fo sufficietly 8 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5
4 Dipty Rai Dhal et al oof The chaacteistics fuctio of But ξ ( ) is ep C t C X 4 υ(4) X 4 υ(4) Stog Result fo Level Cossigs of Rado olyoials / / / ξ / υ(4 ) (4 ) 4 υ(4 ) 4 (4 ) (4 ) υ(4 ) / [LOG (4) (4) [LOG (4) (4) 4 υ(4 ) 4 4 / / 4 / / υ(4 ) Sice υ(4 (4 ) (4 ) 4 υ(4 ) ) 4 log(4 ) (4) 4 υ(4 )(4 We have log(4 ) (4) 4 υ(4 ) 4 ) 4 Hece usig (6) we obtai 4 ep (4 ) CAe ep (4 ) as equied Lea B( ) ξ set of easue at ost ξ C ( ) / This follows diectly fo lea Now 4 υ(4) ecept fo a Ae B 6 6 υ (4 ) 6 Ae B / / / Ae B The last two steps above follow fo () ad (6) Hece by usig leas ad we have R < fo evey sufficietly lage ecept fo a set of easue at ost μep μ' (4 ) μep ( ) Thus we have R adr fo = + k whee =[k/]+ μ ep The easue of the eceptioal set is at ost μ ep μ' (4 μ ep ( ) ) / μ' μ ( ) (7) 4 We defie the evets E ad F as follows: E ={U U + <- + } F ={U < U } μ' 9 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5
5 Dipty Rai Dhal et al Stog Result fo Level Cossigs of Rado olyoials We have show ealie that (E F ) δ Let η be a ado vaiable such that it takes value o E Uf ad zeo elsewhee I othe wods η with pobabilityδ with pobability - δ Let η ae thus idepedet ado vaiables with E (η )=δ ad (η )= δ δ < We wite S if R ad R othewise REFERENCES [] JE Littlewood ad AC Offod O the ube of eal oots of a ado algebaic equatio II oc Cab hilos Soc 5(99) -48 [] J Hajek ad A Reye A geealizatio of a iequality of Kologoov Acta ath Acad Sci Hugay 6(955) 8-8 [] GSaal O the ube of a ado algebaicequatio oc Cab hilos Soc 58(96) 4-44 [4] EA Evas O the ube of a ado algebaicequatio oc Lodo ath Soc (5)(965) [5] GSaal ad D atihai Stog esult fo eal zeos of ado polyoials JIdia ath Soc 4(976) -4 [6] GSaal ad D atihai Stog esult fo eal zeos of ado polyoials II JIdia ath Soc 4(977) 95-4 [7] N Ragaatha ad Sabadha O the lowe boud of the ube of eal oots of a ado algebaic equatio Idia ue Appl ath (98) [8] NNNayak ad S ohaty O the lowe boud of the ube of eal zeos of a ado algebaic polyoial J Idia ath Soc 49(985) 7-5 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5
Strong Result for Level Crossings of Random Polynomials
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