A Note on the Almost Sure Central Limit Theorem in the Joint Version for the Maxima and Partial Sums of Certain Stationary Gaussian Sequences *

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1 Appe Matheatcs Pubshe Oe Jue 0 ScRes A Note o the Aost Sure Cetra Lt Theore the Jot Verso for the Maxa a Parta Sus of Certa Statoary Gaussa Sequeces * Yuafag Wag Quyg Wu Coege of Scece Gu Uversty of Techoogy Gu Cha Ea: wagyuafag89@6co wqy666@guteuc Receve 9 Apr 0; revse 9 May 0; accepte 5 Jue 0 Copyrght 0 by authors a Scetfc Research Pubshg Ic Ths wor s cese uer the Creatve Coos Attrbuto Iteratoa Lcese (CC BY) Abstract Coserg a sequece of staarze statoary Gaussa rao varabes a uversa resut the aost sure cetra t theore for axa a parta su s estabshe Our resut geerazes a proves that o the aost sure cetra t theory prevousy obtae by Marc uzs [] Our resut reaches the opta for Keywors Aost Sure Cetra Lt Theore Statoary Gaussa Sequece Sowy Varyg Fuctos at Ifty Itroucto I ths paper we et ( X X ) be a staarze statoary Gaussa sequece aso et M : ax X M : ax X + be soe sequece of weghts I a () to fucto respectvey S : X : Var ( S ) How to cte ths paper: Wag YF a Wu QY (0) A Note o the Aost Sure Cetra Lt Theore the Jot Verso for the Maxa a Parta Sus of Certa Statoary Gaussa Sequeces Appe Matheatcs { } Φ eote the cator fucto a the staar ora strbu- * Supporte by the Natoa Natura Scece Fouato of Cha (609) a Iovato Project of Guagx Grauate Eucato (YCSZ058)

2 Y F Wag Q Y Wu The ASCLT has bee frst trouce epeety by Brosaer [] a Schatte [] for parta su sce the the cocept has areay starte to receve appcatos ay fes For exape Fahrer a Satuer [] a Cheg et a [5] extee ths aost sure cetra t theore for parta sus to the case of axa of rao varabes Uer soe cotos they prove as foows: ( a ( S b) x) G( x) Ι as for a x R where a > 0 a b R a S b G where G(:) s soe o-egeerate strbuto fucto Afterwars Marc uzs [5] showe the ASCLT ts two-esoa verso e satsfy S I a ( M b) x y e Φ( y) τ as for xy () ( ) exp I ths paper we exte the weght to the weght 0 < X X s a staarze statoary Gaussa sequece the covarace N Our purpose s to prove that f fucto r( t ) fufs r t > 0 a t () t where s a postve sowy varyg fucto at fty Moreover f the uerca sequece { } fes the foowg reato the we have ( ) u sats- Φ u τ for soe 0 τ < () S I a ( M b) x y e ( y) τ Φ as for ay xy R () (( ) ) exp where a 0 < a I the foowg eote by a b f b as by a b f there exsts a costat c > 0 such that a cb for suffcety arge The c stas for a costat whch ay vary fro oe e to aother Ma Resuts Theore Assue that ( X X ) to r( t ) satsfes () for soe > 0 If the uerca sequece { } exp ( ) 0 < be a postve o-ecreasg sequece wth Let { } suabe to a fte t x f where be a staarze statoary Gaussa sequece the covarace fuc- u satsfes () the () hos wth We say that { } x s - x x () 599

3 Y F Wag Q Y Wu Rear By a cassca theore of Hary (see Charasehara a Mashsuara [6]) f two se- * queces a * O the -suabty pes the -suabty e f a se- quece { x } N hos wth the weght { } f ASCLT hos wth { } satsfy s -suabe to x the t s aso the for 0 the for 0 * -suabe to x These resuts show that f () () aso hos wth the weght { } ASCLT aso hos wth { } So So by ths f we use arger weghts we shou expect to get stroger resuts Theore reas va f we exte the weghts fro to (( ) ) exp 0 < < Whe 0 the weghts Lea Uer the sae assuptos as Theore f the uerca sequece { u } fufs () the there exsts soe > 0 a for ay xy a < x x where u + b u + b a a Proofs Proof of Theore S S E IM u IM u () S S Cov IM u IM u () Uer the assuptos of Theore o we have P( M u ) X X r( t ) a { } u by Theore Leabetter et a [7] τ e for τ whch s efe () Let y be a rea uber for each Y eotes a staar ora varabe whch s epeet of ( X X ) a has the sae strbuto as S 0 as S Fro the proof of Lea we get that PM u y P M u P Y y Thus we have by Toeptz Lea we obta where a To prove Theore for S τ PM u P( M u) P( Y ) e Φ( y) S P M u y e ( y) τ Φ () (( ) ) exp 0 < < 0 < < We shou prove the foowg: S S IM u PM u 0 as () I orer to prove () t suffces to show the foowg hos (see Lea Csa a Gochgaza 600

4 Y F Wag Q Y Wu [8]) for soe > 0 S ( + ) () Var I M u S S Set ξ IM u PM u by Lea we have We ow that S S E( ξξ ) Cov I M u I M u S S S Cov I M u I M u I M u S S + Cov I M u I M u S S EIM u IM u S S + Cov I M u I M u for < Var S I M u E ξ E + : T + T ( ) > ( ) For T we have the foowg estate: T ( ξξ) ( ) ( ) ( ) + By the eeetary cacuato t s easy to see that ( ) ( ) For 0 < < we have : > 0 a + The exp ( ) exp ( ) ( ) λ ( ) > < ( ) + ( ) ( ) ( ) ( ) T The (9) a thus (8) foows fro above estates By usg (7) we copete the proof of Theore 60

5 Y F Wag Q Y Wu Proof of Lea We frst coser that () hos wth soe 0< < Let we have S j j j j + 0 < CovX Cov X X + r j + r j () + for soe 0 < < t t By the appcato of the Karaata s theore (see Meczu [9]) we obta Fro () a () for soe 0< < we have L ( )( ) ( ) () ( ) L S L CovX t t L L Sce L( ) s a sowy varyg fucto at fty for ay > 0 L S CovX for ay > 0 a soe 0< < Hece By () r( t) S 0 < sup CovX 0 as for soe so there exst λ 0 such that 0 thus there exst δ such that t t t So 0 < < () S 0 < sup CovX < λ < for ay > 0 () t> 0 < sup r t δ < (5) Let y be a arbtrary rea uber a < Suppose that Y s a rao varabe whch has the sae S strbuto as but Y s epeet of ( X X ) the we have S S E IM u IM u S S PM u PM u S PM u P M u P Y S + PM u P M u P Y + P( M u) P( M u) : A + A + A 60

6 Y F Wag Q Y Wu By Theore Leabetter et a (98) a () () a (5) we get that S u u ( λ) + ( + λ) (6) A+ A Cov X exp exp As we ow { u } fufs () whch pes that u c πu exp cobe ths a (6) we have We set + > 0 thus + λ Next we estate A u exp u ( + λ ) ( ) ( + λ) ( ) ( + λ) A + A + + λ + λ A+ A for soe ( ) ( ) A P M u P M u ( ) ( + λ ) + λ ( ) ( ) + ( u) ( u) + : B + B + B P M u Φ u + P M u Φ u Let δ s efe as (5) set a a () we have + Φ Φ δ < such that 0< < + (7) + λ (8) 0 + δ + > By Theore Leabetter et ( ) u + δ ( ) + δ L + δ t t t + B + B exp r t + δ + δ sce L( ) a are sowy varyg fuctos at fty we have δ the Meawhe foowg fro the eeetary equaty + L for ay > 0 B+ B for soe > 0 (9) x x 0 x We get B for < (0) By (8) (9) a (0) f () hos wth 0< < the () hos Prove () aso hos wth soe sce r () s postve we get Var ( S ) ths py 60

7 Y F Wag Q Y Wu S 0 < CovX As t t t t s a sowy varyg fucto fty we get for t t ay > 0 We obta that () a () By usg Theore Leabetter et a a by (6)-(0) et A - A be repace by C - C () we aso get C+ C et - be repace by B - B (8) we aso obta that + exp ( ) + u δ + δ t + δ r t for soe > 0 e () we get C at ast we obta whe () hos wth the () aso hos Next we prove () Frst we coser the stuato of soe 0< < Let + Sce s a postve sowy varyg fucto the we have ( ) L for ay > 0 s ootoc ecreasg t so by () a () we obta t t t t S 0 < CovX t L t t L t L L t L L t for soe 0 < η < η the there exst ubers µ 0 such that Coparso Lea we obta S S Cov I M u I M u + ( r( j ) ) t S 0 < sup CovX < µ < for a > 0 By the Nora + S S S S PM u M u PM u PM u r j exp + CovX exp j Cov X exp Cov + S + Cov X : E + E + E + E u + u S u + S + Cov X S u S S + + () 60

8 Y F Wag Q Y Wu By (5) set δ < < the 0 + δ + > we have a u + u u + u E r( j) exp exp r t j + ( r( j ) ) ( δ ) + + t + u + u u + u exp ( δ) ( δ) exp + t + t + t + δ δ L + + δ By () set λ < < the E + + < 0 we have + λ S u CovX exp S + Cov X + λ Cov X u S exp ( + λ) + + λ + t for soe 0 < < ( ) ( + λ ) ( ) ( λ ) for 0 < < < S CovX r( j) r ( j ) + + j j + r r + u S E exp Cov X ( µ ) + + u L exp ( µ ) µ ( ) ( ) ( + µ ) L L for 0 < < S S+ S S S S+ S E Cov + E E S S S+ S E E : F + F + F L () 605

9 Y F Wag Q Y Wu Sce s ootoc ecreasg so for t a the property of the sowy varyg fucto we obta By Var ( S ) r( j ) r( j ) + + j j + j + j + L L L by () Karaata s theore F Cov X X r j + r( j) r the we get S S S L L S E + thus E( S S) E( S) euce that for 0 < < < by the statoary of the sequece ( X X ) we get + a by the fact that EX X > 0 for a j a () we L L S S+ S S S S+ S F E E E E where the seco equaty foows fro Jese equaty For F we have S S S+ S F E E 0 so we prove () for soe 0< < Next we prove () for soe > j for soe < S S Cov I M u I M u : G+ G + G + G where G - G are efe as E - E () but for () hos for > Sary as the proof of E - E t s easy to chec that G+ G + G for soe > 0 efe G : H+ H + H as () by Karaata s theore we get By the efto of + + H r j r t ( ) j + t + + r( t) L ( ) for 0 < < t + a Karaata s theore t s easy to obta that t r t t r t c t t t t t t t t 606

10 Y F Wag Q Y Wu the we have ( O ) Sary as the proof of F H for 0 < < by the statoary of the sequece ( X X ) t s easy to see that S S S+ S H E E So we have prove () for the case of > Fay we shou prove () for 0 S S Cov I M u I M u : I+ I + I + I where we repace E - E by I - I as () but for () hos for soe Sary t s easy to chec that I + I + I for soe > 0 efe I : J+ J + J Sce r( t ) s ootoc ecreasg efe L r( t) ˆ + by the appcato of the proposto (Potter s TH) t of the sowy varyg fucto for ay 0 < δ < we have J J + + r t + + r t ( ˆ ) ( Lˆ ( ) ) ( ˆ ) ( Lˆ ) δ δ ( + ) t + t L L Lˆ ( ˆ ) ˆ L L δ δ + for soe 0 < < ˆ δ ˆ L ˆ ˆ L for soe 0 < < Sary as F 0 whe 0< < we have J 0 So we have I Fay we get that I+ I + I + I for soe > 0 whe Refereces [] uzs M (008) The Aost Sure Cetra Lt Theores The Jot Verso for The Maxa a Sus of Certa Statoary Gaussa Sequece Statstcs & Probabty Letters [] Brosaer GA (988) A Aost Everywhere Cetra Lt Theore Matheatca Proceegs of the Cabrge Phosophca Socety [] Schatte P (988) O Strog Versos of the Cetra Lt Theore Matheatsche Nachrchte [] Fahrer I a Statuer U (998) O Aost Sure Cetra Max-Lt Theores Statstcs & Probabty Letters

11 Y F Wag Q Y Wu [5] Cheg H Peg L a Q YC (998) Aost Sure Covergece Extree Vaue Theory Matheatsche Nachrchte [6] Charasehara K a Mashsuara S (95) Typca Meas Oxfor Uversty Press Oxfor [7] Leabetter MR Lgre G a Rootze H (98) Extrees a Reate Propertes of Rao Sequeces a Processes Sprger New Yor [8] Csa E a Gochgaza K (00) Aost Sure Lt Theores for The Maxu of Statoary Gaussa Sequeces Statstcs & Probabty Letters [9] Meczu J (00) Soe Rears o the Aost Sure Cetra Lt Theore for epeet Sequeces Lt Theores Probabty a Statstcs