STRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM

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1 Joural of Statitic: Advace i Theory ad Applicatio Volume, Number, 9, Page STRONG DEVIATION THEORES FOR THE SEQUENCE OF CONTINUOUS RANDO VARIABLES AND THE APPROACH OF LAPLACE TRANSFOR School of athematic ad Iformatio Sciece Shadog Ititute of Buie ad Techology Yatai 645 P. R. Chia wagxuewuxx@63.com Abtract We ue the otio of the log-lielihood ratio to tudy ome limit propertie of the equece of oegative cotiuou radom variable i more geeral coditio. A cla of trog deviatio theorem repreeted by iequalitie are obtaied. The reult preeted i the paper exted ad improve the mai reult of Liu, Wag ad Li.. Itroductio I recet year, ome reult have bee obtaied i the field of the deviatio for the arithmetic mea of the radom variable. I particular, the trog deviatio theory for the dicrete radom variable ha bee tudied exteively by author Liu [3] ad Wag [5] ad have bee obtaied better reult. The mai problem, tracig bac to Liu [4] i to determie a relatiohip betwee the true probability ditributio ad it athematic Subject Claificatio: 6F, 6F5. Keyword ad phrae: log-lielihood ratio, trog deviatio, Laplace traform, cotiuou radom variable. Received arch 5, 9 9 Scietific Advace Publiher

2 36 idepedet product ditributio. Liu [] dicued the trog deviatio theorem for the equece of oegative idetical cotiuou radom variable. Wag [6] tudied arbitrary tochatically domiated cotiuou radom variable ad obtaied ome deviatio theorem ad ome trog law of large umber. Uder the coditio that the tail probability fuctio exit a upper boud fuctio, Li et al. [] provide a lower boud for lim if ( X EX ) ad a upper boud for = lim up ( ) X EX i term of ome fuctio of the = Laplace traform of the tail for oegative cotiuou radom variable X ( =,, ). The mai purpoe of thi paper i to etablih the trog deviatio theorem repreeted i the form of the iequality for the equece of oegative cotiuou radom variable, by dicuig the uperior limit ad the iferior limit of ( X EX ) uderlyig the weaer = coditio tha the literature [,, 6]. oreover, we obtai the X = etimatio of the deviatio betwee ad EX. = Let { X, } be a equece of oegative itegrable radom variable o the probability pace ( Ω, F, P ). Aume that the joit ditributio deity fuctio of the equece { X, } i f ( x, x,, x ), x,, =,,. () Let f ( x )( =,, ) tad for the ditributio deity fuctio of the radom variable X, repectively. Deote the product ditributio Write π ( x, x,, x ) = f ( x ). () = r ( w) = f ( X, X,, X ) π ( X, X,, X ) = f ( X, X,, X ) ; f ( X ) = (3)

3 STRONG DEVIATION THEORES FOR THE SEQUENCE 37 L ( w) = l r ( w) ; = lim up L ( w), (4) r ( w) ad L ( w) are the lielihood ratio ad the log-lielihood ratio, repectively. Here, a uual, the radom variable r ( w) i the ample relative etropy rate. Obviouly, if { X, } are idepedet, the f ( X, X,, X ) = f ( X ), ad r = ( w) =. Defitio. Let { X, } be a equece of radom variable, ad f ( x ), =,,, be the margial deity fuctio of f ( x, x,, x )( =,, ). Let Laplace traform ad the tail probability Laplace traform be a follow: ad x f () = e f ( x ) dx, (5) x q () = e q ( x) dx, (6) q ( ) = ( ). >. x f x dx x (7) x Throughout thi paper, we aume that there exit (, ) uch that Sice f () <, q () <, [, ], =,,. (8) e x x x x, x >, N. (9)! It follow from (5), (8) ad (9) that <,, N. EX

4 38 Lemma []. Let f (), q () be defied by (5) ad (6), repectively. The we have q () = [ f ()]. Lemma [, 6]. Let f ( x, x,, x ) ad g ( x, x,, x ) be two probability deity fuctio o ( Ω, F, P ). Suppoe that t ( w) = g ( x, x,, x ) f ( x, x,, x ), the lim up l t ( w), a... (). ai Reult ad Proof Theorem. Let { X, } be a equece of oegative cotiuou radom variable o the probability pace ( Ω, F, P )., f (), q () be defied a above (4), (5) ad (6), D = { w : < } ad P ( D ) =. The lim up = ( X EX ) α( ), a.., () α( x ) = if{ ϕ(, x) : < }, x, () ( ) [ () x ϕ, x = lim up q q ()], x, <, = (3) ad α( x), lim α( x) = α( ) =. x (4) Proof. For arbitrary [, ], let x g (, x ) = e f ( x ) f (), x, N. (5) Obviouly, g (, x) dx =. Suppoe further that

5 STRONG DEVIATION THEORES FOR THE SEQUENCE 39 q x = (6) = = (, x,, x ) = g (, x ) [ e f ( x ) f ()], the q (, x,, ) i a multivariate probability deity fuctio. We aume that x t q (, X,, X ) (, w) =, (7) f ( X,, X ) X = X ( w), w Ω. I view of Lemma, there exit D () F, P ( D() ) = uch that lim up l t (, w), w D( ). (8) From (6), (7) ad (8), we have lim up { X l f ( ) l r ( w)}, w D( ). = = (9) By (4) ad (9), we get lim up = X lim up = l f (), w D(). () Settig = i (), we obtai, w D( ). () Let, dividig the two ided of () by, we get < lim up = X lim up = l f (), w D(). () By (), the property of the uperior limit: lim up ( a b ) d lim up ( a c ) lim up ( b c ) d ad the iequality l x x ( x > ), ad Lemma, for w D( ), we have

6 4 lim up = [ X EX ] lim up = lim up = l f () EX f () EX lim up = [ ( ) () r w q q ()]. (3) Let Q be the et of ratioal umber i the iterval [, ), ad let D = D(), the P ( D) =. We have Q lim up = [ X EX ] lim up [ q () = q ( )], w D, Q. (4) Let g() = lim up [ () q q ()], <. = (5) By virtue of (), (3) ad (5), we get x α( x ) = if{ g( ) : < }, x ; (6) x ϕ(, x) = g( ), x, <. (7) Clearly, g () ad ϕ(, x). Thi implie α( x ). Now, we prove that g () i a cotiuou fuctio i iterval [, ]. I fact, for ay ε >, if <, it follow from (8) that for each, there exit a cotat > uch that ( ) x e q ( x) dx < ε. Sice

7 STRONG DEVIATION THEORES FOR THE SEQUENCE 4 x e q ( ) ( ) x x ( ) [ ( ) x x dx e q x dx e e ] q ( x) dx ε x ( e ) e q ( x) dx ε ( e ) q ( ) ε. (8) It i eay to ee that atifyig whe. Thi implie that ( ) x e q ( x) dx < ε i o icreae x ( ) ( ) x e q x dx e q ( x) dx ε, a, N. (9) Therefore, it follow from (5) ad (9) that > g() g( ) = lim up [ q () ()] lim up [ ( ) q q q ( )] lim if lim if = lim if = = = = [ q () q ()] lim if [ q ( ) q ( )] [ q () q ( )] x e q = = ( ) ( ) x x dx e q ( x) dx ε, a. (3) By (3), we ow that g () i a cotiuou fuctio with repect to o the iterval [, ]. Hece, ϕ (, x) i alo a cotiuou fuctio with repect to o the iterval [, ). From (6), for each w D D, tae ( ) w Q, =,,, uch that

8 4 lim ϕ( ( w), ) = α( ). (3) From (4)-(7) ad (3), we obtai lim up ( X EX ) ϕ( ( w), ), w D D. = (3) We alo have lim up ( X EX ) α( ), w D D. = (33) By P( D D ) = ad (33), () hold. Followig, we prove that lim α( x) =. x Whe x [, ), taig = x, It follow from (6) that α( x) g( x ) x. (34) By virtue of the cotiuou property of g ( ) o iterval [, ] ad g ( ) =, we have lim α( x) = α( ) =. x (35) The proof of Theorem i completed. Imitatig the proof of Theorem i thi paper ad the proof of Theorem i [], we ca etablih the followig theorem: Theorem. Let { X, } be a equece of oegative cotiuou radom variable o the probability pace ( Ω, F, P )., f (), q () be defied a above (4), (5) ad (6), D = { w : < } ad P ( D ) =. The lim if = ( X EX ) β( ), a.., (36)

9 STRONG DEVIATION THEORES FOR THE SEQUENCE 43 β( x ) = up{ v/ (, x) : < }, x, (37) ( ) [ () x v/, x = lim if q q ()], x, <, = (38) ad β( x), lim β( x) = β( ) =. x (39) Theorem 3. Let { X, } be a equece of oegative cotiuou radom variable o the probability pace ( Ω, F, P )., f () be defied a above (4) ad (5), D = { w : < } ad P ( D ) =. Let m = lim if EX. The = lim if ( Xi EXi ) ( m, ), a.., δ i= (4) m δ ( m, ) = up{ : < }, (4) ad δ( m, ), lim δ( m, ) =. Proof. By the proof of Theorem, () hold, that i lim up X = lim up l f (), w D(). = (4) Let <, dividig the two ided of (4) by, we get lim if = X lim if = l f (), w D(). (43) By (43), the property of the iferior limit: lim if ( a b ) d lim if ( a c ) lim if ( b c ) d we have

10 44 lim if = = [ X EX ] lim if [ l f () EX ], w D(). (44) By e x x x a x < ad l ( x) x a x >, we obtai X l f () = l Ee l E( X ( X ) ) EX EX. (45) It follow from (44) ad (45) that lim if = [ X EX ] lim if EX =, w D(). (46) Let ( ) (, ( )) up m r w δ m r w = : <, (47) m = lim if EX. (48) = Obviouly, δ( m, ). we tae = whe (, ]. We obtai Therefore, (, m m δ m ) = ( ) r( ). w (49) lim δ( m, ) =. ( w) (, ], =,,, uch that From (47), for each w D, tae lim ( m ) = δ ( m, r ( w )). (5) Let D = D( ), the P ( D) =. We have by (46) N

11 STRONG DEVIATION THEORES FOR THE SEQUENCE 45 m lim if [ X EX ], w D D, (, ]. (5) = Sice P( D D ) =, it follow from (47), (5) ad (5) that (4) hold. Thi complete the proof of Theorem 3. Theorem 4. Let { X, } be a equece of oegative cotiuou radom variable o the probability pace ( Ω, F, P ). There exit a cotat uch that X ( w), for all N ad w D ad P( D ) =., f ( ) be defied a above (4) ad (5), D = { w : < } ad P ( D ) =. Let m = lim up EX. = The lim up = ( X EX ) (, m), a.., (5) em ( m, ) = if{ : max{, } }, (53) ad ( m, ), lim ( m, ) =. Proof. By the proof of Theorem, () hold, that i lim up = X lim up = l f (), w D(). (54) Let, dividig the two ided of (54) by, we get < lim up X = lim up = l f (), w D(). (55) By (55), the property of the uperior limit: lim up ( a b ) d lim up ( a c ) lim up ( b c ) d, we have

12 46 lim up = By = [ X EX ] lim up [ e x x ex l f () EX ], w D(). (56) a x ad l ( x) x a x >, we obtai whe X. X e l f () = l Ee l E( X e( X ) ) EX EX. (57) whe [ max{, }, ), it follow from (56) ad (57) that e lim up [ X EX ] lim up EX, w D() D. = Let ( ) (, ( )) if em r w m r w = : max, <, (59) m = lim up EX. (6) = Obviouly, ( m, ). By puttig = whe r ( w) [ max{, }, ), we obtai (, em em m ) = ( ), w D. (6) Therefore, lim we have by (58) ad (59) = (58) δ( m, ) =. Similar to the proof of Theorem 3, lim up ( X EX ) ( m, ), a... = (6) Thi complete the proof of Theorem 4.

13 STRONG DEVIATION THEORES FOR THE SEQUENCE 47 Remar. Theorem (-4) how that the maller r ( w) i, the maller the deviatio i. Therefore, r ( w) meaure the deviatio betwee the reality ditributio f ( x,, x ) ad the idepedet product ditributio f ( ). x = Remar. Theorem ad i thi paper remove the coditio that the tail probability fuctio exit a upper boud fuctio i Theorem (-4) i [] ad drop the coditio that the equece exit a tochatically domiated by a oegative radom variable i Theorem (-4) i [6]. oreover, we exted the mai reult i [] to the oegative cotiuou radom variable with differet ditributio. Remar 3. Theorem 3 ad 4 i thi paper ot oly how that the maller r ( w) i, the maller the deviatio i, but alo obtai the deviatio etimatio betwee X = ad EX. = Referece [] G. Li, S. Che ad J. Zhag, A cla of radom deviatio theorem ad the approach of Laplace traform, Stat. Probab. Lett. 79 (9), -. [] W. Liu, A cla of trog deviatio theorem ad the approach of Laplace traform, Chiee Sciece Bulleti 43 (998), [3] W. Liu, A cla of mall deviatio theorem for the equece of oegative iteger valued radom variable, Taiwaee J. ath. (997), 9-3. [4] W. Liu, Relative etropy deitie ad a cla of limit theorem of the equece of m- valued radom variable, A. Probab. 8 (99), [5] X. W. Wag, A cla of trog deviatio theorem for the equece of oegative iteger valued radom variable, Stat. Probab. Lett. 78 (8), [6] Z. Z. Wag, A cla of radom deviatio theorem for um of oegative tochatic equece ad trog law of large umber, Stat. Probab. Lett. 76 (6), 7-6. g