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1 Poof of Theoem Sulemetal Mateial Simila to the oof of Theoem, we coide the evet E, E, ad E 3 eaately. By homogeeity ad ymmety, P (E )=P (E 3 ). The aoximatio of P (E ) ad P (E 3 ) ae idetical to thoe obtaied i Sectio 3. by ettig (x). Theefoe, P (E )=P (E 3 ) D h u h e u h /. Fom the deivatio of P (E ) i the eviou oof, we obtai that P (E \ E 3 )=o(p (E )). Fo the et of the oof, we how that P (E )=o(p (E )) ad thu P (E \ E )=o(p (E )). oximatio of P (E ). et H(x, u) be a defied fo Theoem ad u olve H( (u),u)e u = b, whee (u) =u / / /. Fo the et of the oof, we will how that P (E )=O()e u O(u"). (33) fo ay ">. ccodig to the dicuio i Sectio, thee exit a " > uch that u>u h " ad thu e u O(u") = o()u h e u h /. If the above boud i (33) ca be etablihed, the we ca coclude the oof. Fit, we deive a aoximatio fo (u, ") =P max v (x) >b, x[ u /", u /" ] whee ">ichoe mall eough. The, we lit the egio [,] iton = u /" may iteval each of which i a locatio hift of [, u /" ], i.e. [ku /", ku /" u /" ]. Thak to the homogeeity of (x), the aoximatio fo P max x[ku /",ku /" u /" ] v (x) >b ae the ame fo all ale k ale N. The, we have P [ N k= { max v (x) >b} ale ( o()) x[ku /",ku /" u /" ] u (u, "). /" I what follow, we deive a aoximatio fo (u, "). The deivatio i imila to the oof of the Theoem. Theefoe, we omit the detail ad oly lay out the key te ad the majo di eece. We exad (x) aoud x = coditioal o (by edefiig the otatio) ( )=u w, ( )=y, ( )= (u ) 5

2 ad obtai that (x) = y u w (u ) y 6 x3 u 4 x4 g(x (u ) Similaly, we have the followig ooitio fo localiatio. x ) (x ). y (u ) Pooitio 4 Fo > 3", let G u = { w >u 3" } [ { y >u /4" } [ { >u /4" } o [ u g(x) ux > [ u g(x) >u x/[ u /",u /" ] x[ u /",u /" ] / o Ude the coditio of Theoem, we have et P (G u ; We ow oceed to the facto max x[ u /", u /" ] F (x) u = G c u. R v (x) >b)=o()e u /. F (t)e (t) dt R e. (t) dt Followig exactly the ame deivatio a emma i Sectio 3. ad otig that (x),we have that R F (x) dt R e = ex (t) dt 3 4 (u ) 3 o(u )!(u), whee we edefie a chage of vaiable = x y (u ). Thu, imila to (9), we obtai that v (x) = e (x) "F (t) = e u w y ex R (u ) e F (t)e (t) dt R e (t) dt u # 6 y( y (u ) )3 u 4 ( y (u ) )4 o 3 4 (u ) 3 o(u )!(u). 6

3 We futhe imlify the above dilay ad obai that v (x) = e u w y ex u (u ) e 4 u y 8 4 u 3 y4 y 3h i 3 3 u 3 4 u o 3 O(u )!(u). Fo all y ale( " ) u /", we have that max v (x) ale max v (x) x[ u /", u /" ] x[ (" )u /", (" )u /" ] = e u w y (u ) e ex That i, v (x) i maximied whe x = / / u /,the Thu, we have that imlie that u 4 u y 8 4 u 3 y4 y 3h i 3 3 u 3 4 u 3 o O(u u )!(u). (34) 3 3 u 3 4 u 3 y (u ) o(u )O( /u). Sice = =. max v (x) > b x[ u /", u /" ], w!(u). y (u ) u 4 3 u y 8 4 u 3 y4 O( u )O(u ) Coeodig to the aalyi i Sectio 3., the ext te i to iet to S(w, y, ) ad obtai 7

4 that S(w, y, ) = u w (w ) u(w y u ) = u ( w / ) u y u y 3 u y u O( /u)o() = u ( w / ) u y u y 4 u = u ( w / ) u h y u i 4 O(u8" ). 4 O( /u)o() Fo the lat te i the above deivatio, we ue the fact that, o the et u, O( /u) =O(u 8" ). Thu, P max v (x) >b x[ u /",u /" ] = h(w, y, )P ( max v (x) >b w, y, )dwdyd u x[ u /",u /" ] = O()e u O(u8" ) 8 u P ( >!(u)) ex We itoduce a chage of vaiable u ( w / ) h y u i o dwdyd. B = ( y u ). The, w / = B w ( y u ) = B o(). Thu, by domiated covegece theoem ad alyig the chage of vaiable fom (w,, y) to (,B,y), we have that! P max v (x) >b; y ale( " ) u /" ; u = O()e u O(u8"). (35) x[ u /", u /" ] Fo y > ( " ) u /", ote that the fuctio v (x) i maximied at x = y (u ), 8

5 that i outide the iteval [ u /", u /" ]. Theefoe, max x[ u /", u /" ] v (x) i le tha the etimate i (34) by at leat a facto of e u " (by coideig the domiatig tem u e ). Theefoe, max v (x) >b x[ u /", u /" ] if = w y (u ) u 4 3 u y 8 4 u 3 y4 O( /u )O(u ) > u "!(u). Thu, P max v (x) >b;( " ) u /" ale y aleu /4" ; u x[ u /", u /" ] = O()e u O(u8"). (36) We combie the olutio of (35), (36), emma 4 ad obtai that Thu (u, ") = P max v (x) >b x[ u /", u /" ] = O()e u O(u8"). P (E )=O()u / " (u, ") =O()e u O(u8"). " ca be choe abitaily mall, we obtai (33) by edefiig ".! B Poof of Pooitio Poof of Pooitio. The oof eed a chage of meaue decibed a follow. Fo R, let = {x : (x) > }\[x u / /, u / ] be the excuio et (o the iteval [x u / /, u / ]) ove level ad let P be the udelyig omial (oigial) obability meaue. Defie Q ( ) via dq = me( ) E(me( )) dp = me( ) R dp, (37) u / x u P ( (x) > )dx / / whee E( ) i the exectatio ude P ad me( ) i the ebegue meaue of the excuio et above level. Note that ude Q, almot uely u (x) >. I ode to geeate amle ath accodig Q, oe fit imulate with deity fuctio h ( ) : [x u / /, u / ] h( ) = P ( ( ) >b) E(me( )) (38) that i a uifom ditibutio ove the iteval [x u / /, u / ]; the imulate ( ) coditioal ditibutio (ude the oigial law) give that ( ) > ; latly imulate { (x) : x 6= } 9

6 give (, ( )) accodig to the oigial ditibutio. If i uitably choe, Q eve a a good aoximatio of the coditioal ditibutio of (x) give that u x[xu / /, u / ] (x) > b. emma Ude coditio i Theoem, we have that P u (x) >u (log u),e = o(u e u / ). x[x u / /, u / ] Poof of emma. et F b = { u x[x u / /, u / ] (x) >u (log u) }. et = u (log u) /u. The, the obability ca be witte a ale P ( >u (log u) P (F b,e ) ale O()E Q ) ; F b,e me( ) u / ale P ( >u (log u) = O() E Q ) ; F b,e b d, x u / / me( ) whee we ue E Q to deote the coditioal exectatio E Q ( ) ude the meaue Q. Give a aticula [x u / /, u / ], we edefie the chage of vaiable ( ) =u w, ( ) =y, ( ) = (u ). Note that the cuet defiitio of (w, y, ) i di eet fom that i the ooitio ad Theoem. the eviou defiitio of (w, y, ) will ot be ued i thi lemma, to imlify the otatio, we do ot ceate aothe otatio ad ue (w, y, ) di eetly. Coditioal o (w, y, ) the oce g(x) i a mea eo Gauia oce uch that (x) =E( (x) w, y, )g(x ). We have the boud of the excuio et that E Q (/me( )) = O(u), the detailed develomet of which i omitted. With thi i mid, we fit have that that ad imilaly ale P ( >u (log u) E Q ) ; u / /6,F b,e b = o(u e u / ). me( ) ale P ( >u (log u) E Q ) ; y u / /6,F b,e b = o(u e u / ). me( ) I additio, fo ome u cietly lage ad mall, we have that! E P ( >u (log u) ) ; u g(x) > u 4, o u g(x) ux > me( ) x aleu / x >u / = o(u e u / ).

7 The, we oly eed to coide the ituatio that y <u / /6 ad <u / /6. Futhemoe, uig Taylo exaio o (x) a we had doe eveal time eviouly, the oce (x) ia aoximately a quadatic fuctio with mode beig y (u ) fo [x u / /, u / ]. Thu, whe coideig the itegal R e (t) dt ad R (F (x) F (t))e (t) dt, we do ot have to coide the bouday iue a i the aalyi of P (E ). With the ame calculatio fo () by exadig at itead of x, we obtai that if ad oly if = w u v (x) b x[u /, u / ] y (u ) 6 y( (x) (x) ( 4 (u ) 3 o(u )!(u), y (u ) )3 u 4 ( (u ) ) (x) 6(x) ( log (x) (x ) y (u ) )4 3 (u ) ) whee the x i (x) i x = y (u ) o(u )O( /u). Simila to the deivatio fo (39), we exad the ecod ow i the defiitio of ad obtai that = w y u u y (x) (x) ( y (u ) 3 (u ) ) (x ) 6(x ) ( y 4 (u ) 3 (x) ) log (u ) (x ). (u ) 4 4 Notice that Whe x (x) 6(x) ( x <", by Taylo exaio 3 (u ) )=O(u ). (x) ( (x) (u ) ) = O((x x )/ u)=o(log (x) log (x )); whe x x >" (x) ( (x) (u ) ) = O(u / )=o() = o(log (x) log (x )). (x) Theefoe (x) ( (u ) ) i alway of a malle ode tha log (x) log (x ). O the u / / egio x x >, thee exit a oitive uch that log (x) (x ) ale u.

8 Thu, i bouded by < = w y u u y u y (u ) 3 y 4 (u ) (u ) 4 4 Futhemoe, otice that ale P ( >u (log u) ) ale E Q ; y, aleu / /6,F b,e me( ) O() e S(w,y,) P (!(u),f b ) dwdyd. w (log u) me( ) Simila to the eviou develomet, we wite S(w, y, ) = u w = u w h u (w ) u(w y u ) (w ) y u y (u ) 3 y 4 (u ) (u ) 4 4 u / i. Thu, by domiated covegece theoem ad the fact that me( ) = O(u), we have that ale ale ale P ( >u (log u) E Q ) ; y, aleu / /6,F b,e me( ) O() E(me( ) ;!(u))e S(w,y,) dwdyd y, aleu /"/4 O()e u u / E(me( ) ;!(u)) y, aleu /"/4 ale u ex ( ) y u = o(u e u / ). y u y 4 3 u dwdyd With a comletely aalogou oof a the emma, we have that emma Ude coditio i Theoem, we have that P! u (x) >u (log u),e = o(u e u / ). x[u /,x u / / ] We wite J b = { u x[u /,x u / / ] (x) >u (log u) }[{ u (x) >u (log u) } x[ u / /, u / ]

9 ad thu P (J c b,e )=o(u e u / ). We oceed to the followig lemma to comlete the oof of the ooitio. emma et (w, y, ) defied a i Sectio 3.. Fo ">, let Ude coditio of Theoem, we have that b = { w <u 3, y <u /4, <u /4 } P ( c b,jc b,e )=o(u e u / ). Poof. Note that v (x) >bimlie that (x) > log b ale = u O(log u) fo ome ale >. Thu, o the et Jb c, E imlie that u [x u / /,x u / / ] (x) > log b (log u). Theefoe, we have that P ( w >u 3,F c b,e b) ale P ( w >u 3, u [x u / /,x u / / ] (x) > log b (log u) )=o(u e u / ), whee the lat te i a alicatio of Boel-TIS lemma ([6, 5, 4]). Futhemoe, by imly boud of Gauia ditibutio, we have that ad P ( w <u 3, >u /4,F c b,e b)=o(u e u / ), P ( w <u 3, y >u /4,F c b,e b)=o(u e u / ). We thu coclude the oof. The eult of emma,, ad immediately lead to the cocluio of Pooitio. Poof of Pooitio. (w, y, ). Theefoe, P u [ g(x) x >u /8 Note that g(x) i ideedet of (w, y, ) ad u oly deed o ux ] >, u! = P u x >u /8 [ g(x) = o(u e u / ). ux ] >! P ( u ) The lat te i a diect alicatio of the Boel-TIS lemma ad the fact that P ( u )=O(e u /O(u 3 ) ). With a imila agumet, we obtai the ecod boud. Poof of Pooitio 3 ad 4. The oof of thee two ooitio ae comletely aalogou to that of Pooitio, that i, baically a eeated alicatio of Boel-TIS lemma ad the chage of meaue Q. Theefoe, we omit the detail. C Poof of the emma Poof of emma. O the et x x <u /8 ad u,wehave = O(u 8 ) ad thu y 3 (u ) 5/ = O(u ), y (u ) = O(u 4 ), 3 4 (u ) = O(u 3 ).

10 et X be a tadad Gauia adom vaiable. We coclude the oof by the followig calcuatio x x <u /8 e [ = e o(u ) = e o(u ) = = x x <u /8 e E " ex ex 7/ (u ) 5/ y 4 3 (u ) 4 (u ) 4 ] d 7/ (u ) 5/ y 4 3 (u ) # / y 3 X y X 7/ (u ) 5/ 4 3 (u ) X 4 4 (u ) y 4 3 (u ) 8 (u ) o(u ) y 4 3 (u ) 8 u o(u ). 4 (u ) 4 d Poof of emma. We ue the eult of emma ad the Taylo exaio F (x) F (t) =(x)(x t) Recall the chage of vaiable (x)(x t) 6 (x)(x t) 3 o(x t) 4. (t) = y (u ) t x (u ) at the begiig of Ste of the mai oof. We aly it to the atial idex t. Note that t x (t)/ (u ) =y/( (u )) ad x t = (t)/ (u ). We efom the ame littig a i (5), iet the eult i (6), ue the exaio of i (3), ad obtai that e (t) dt (F (x) F (t))e (t) dt y = ex 4 3 (u ) 8 (u )!(u)o(u ) ale (x) aleu 8 (u ) (x) e [ 6 (x) (u ) 3 o(u 3/ ) 3 7/ (u ) 5/ y 4 3 (u ) (u ) 4 (u ) 4 ] d We ewite the above itegal by ullig out the Gauia deity ad exadig the exoetial 4

11 tem i the lat ow y = ex 4 3 (u ) 8 (u )!(u)o(u ) e aleu 8 " (x) ale (u ) (x) (u ) 6 (x) 3 7/ (u ) 5/ y 4 3 (u ) 4 (u ) 4 d. # 3 (u ) Simila to emma, we futhe evaluate the above itegal by comutig momet of N(, / ) ad obtai that (we omit eveal co tem that ca be abobed by o(u )) F (x) = ex ale R F (t)e (t) dt R e (t) dt y 4 3 (u ) 8 (u )!(u)o(u ) (x) (x) (x) (u ) 6 (x) 3 4 (u ) 3 (x) 3 3 (u ) y 4 3 (u ) (x) 8 (u ). We take out the facto (x) fom the backet ad cotiue the calculatio y = ex 4 3 (u ) 8 (u )!(u)o(u ) h (x) ex (x) (x) ( (u ) ) (x) 6(x) ( y 3 4 (u ) (u ) 8 (u ) 3 (u ) ) i. We futhe imlify the above dilay ad obtai that h = (x) ex (x) (x) ( (u ) ) (x) 6(x) ( 3 4 (u ) 3 o(u )!(u) i. 3 (u ) ) Poof of emma 3. et be defied a i (). Note that (x ) = ad (x) (x )( y/ (u )). We aly Taylo exaio of the tem log (x (u ) y) (x ) i () ad exad 5

12 the ecod ow of (). Thu, ca be futhe imlified to = w y u u y y 3 3 (u ) 4 (u ) (x ) (x ) ( y (u ) )( (x ) 6(x ) ( (x ) (x ) ( Note that = u / / /.Thetem y (u ) 3 u 4 4 ) (u ) 3 (u ) ) 3 4 (u ) 3 y (u ) ) o(y u )O( /u ). (x ) (x ) y (u ) ( (u ) ) exaded fom the thid ow cacel the co tem (x ) (x ) y (u ) exaded fom the quadatic tem i the lat ow. The, i futhe imlified to = w y u u y y 3 3 (u ) 4 (u ) (x ) (x ) ( (u ) ) (x ) 6(x ) ( (x ) (x ) ( y (u ) 3 (u ) 4 3 (u ) ) 3 4 (u ) 3 y (u ) )o(y u )O( /u ). Futhemoe, the tem 3 3 (u ) i the ecod ow cacel ow lug i = u ad obtai that = w y u y 4 u y 8 4 u 3 u y 4 3 u (x ) 4 u 3(x ) u (x ) (x ) ( u = w y u u y u 4 u (x ) 6(x ) y 4 8 u 3 y u ( 3 4 (u ) (x ) (x ) )o(u y u )O( /u ). u (39) 4 i the fouth ow. We y u )o(u )O( /u) 6

13 Poof of emma 5. By imle algeba, we have that S(w, y, ) = u u/ ( w / ) y u y 4 4 u y4 u ( 3 (x ) (x ) = u u/ ( w / ) y (x ) 3 u (x ) y 3 u )o(y /u)o( /u)o() y u o(y /u)o( /u)o(). Note that, o the et u, o(y /u)o( /u) =o(y /u ) ad thu, S(w, y, ) u u/ y 3 u o() y 3 u (x ) y (x ) u O(). Poof of emma 6. e (t) dt = e c Uig the ecod chage of vaiable i (4), the deomiato i (5) i h ex 3 7/ u 5/ y 4 3 u 4 4io dt. u et be a tadad Gauia adom vaiable followig N(, ). With a imila littig i (5) ad the deivatio i emma ad oticig the bouday cotait that ( /u ) y t ale () ale (u ), we aly Taylo exaio o the itegad ad have that = = e c o(u ) (u ) e!(u ) " / y E 3 7/ u 5/ 4 3 u 4 4 ; ale u e c o(u ) (u ) e!(u )O(y 3 /u 5/ y /u ) ale E 4 4 ; ale u u (u ) y, u (u ) y # whee c = (u w y (u ) y (u ) 3 ) ad!(u) =O(u x aleu /8 g(x) ). The exectatio 7

14 i the eviou dilay ca be witte a ale E 4 4 ; ale u u (u ) y h i = P ale u (u ) y ex 4 E( 4 ale )!(u )O(y 3 /u 5/ u y /u y/u 3/ ) We ue the fact that E( 4 ale )=E( 4 ale cotiue the calculatio ad obtai that e e (t) c o(u ) dt = h (u ) P ale ex q u u q (u ) y i (u ) y)o( yu / ). We 4 u E( 4 ale )!(u )O(y 3 /u 5/ y /u y/u 3/ ). We ow oceed to the umeato of (5). Uig Taylo exaio F (x) F (t) =(x)(x t) the umeato of (5) i (with the littig a i (5)) (F (x) F (t))e (t) dt = ec!(u )o(u ) (u ) " = (x)( e E (x)( q ( /u) y (u ) u 8 (u ) ) (x)( (x)(x t) 6 (x)(x t) 3 o(x t) 3, (u ) ) 6 (x)( 3 7/ (u ) 5/ y 4 3 (u ) 4 (u ) od 4 (u ) ec!(u )o(u ) (x) 4 u 4 ( (u ) ) (x) ( (u ) ) (x) ( 6 (u ) ) O(y 3 /u 5/ y /u u ) ; ale u (u ) y o. # (u ) )3 o(u 3/ ) (u ) )3 8

15 Thu, the facto i (5) i = ex (F (x) F (t)) E (x)( e (t) R e () d dt 4 u E 4 ale (u )!(u ) (u ) ) (x) ( (u ) ) (x) ( 6 (u ) ) ale (x) 4 u 4 ( o u (u ) )3 (u ) y o whee (u )=O(y 3 /u 5/ y /u y/u3/ )o(u u ). We take out a facto (u ) fom the above exectatio ad obtai that o = ex 4 E 4 ale (u )!(u ) u (x)( u ( /u ) E (x) (u ) ) ( (u ) ) u (x) 6 u ( u ) 3 (x) 4 u 4 ( u ) ale u (u ) y o. Notice that i the lat two tem of the above dilay ad fo the deomiato of the ecod tem i the ecod low, u i elaced by u. The eo caued by thi chage ca be abobed ito (u ). Notice that u ( /u ) = e o(/u u ). u We futhe eaate the exectatio ito two at ad obtai that = ex E o 4 E 4 ale (u )!(u ) u h (x)( (u ) ) (x) ( (u ) ) ale u h (x) E ( u ) 3 6 u Thu, we coclude the oof. (x) 4 u 4 ( u ) ale e u u u u (u ) y i (u ) y io. 9

16 Poof of emma 7. Simila to the calculatio eultig (8), we obtai that y (x) = u w (u ) g(x t )#(x t ) = u w y (u ) u (u ) u 4 6 y( y (u ) )3 u 4 ( y (u ) )4 4 o(u y )g(x t )#(x t ), whee #(x) =O(u /4 x 5 ux 6 ). Combiig the above exeio ad emma 6, we obtai that v (x) = e (x) e (t) (F (x) F (t)) R e () d dt = ex E (u )O(y u )!(u ) u w y ex u (u ) h (x)( (u ) ) u (x) ( (u ) ) ale u h (x) E ( u ) 3 6 u (x) 4 u 4 ( u ) ale 4 o u u 4 o 4 E 4 ale u Uig Taylo exaio o the two exectatio tem, we obtai that h E (x)( (u ) ) (x) ( (u ) ) ale u h (x) E ( u ) 3 6 u (x) 4 u 4 ( u ) ale h = E (x)( (u ) ) u u u u (u ) y i (u ) y i (x) ( (u ) ) i ale u u (u ) y 8 h < E (x) 6 u ( u ) 3 (x) 4 ex u 4 ( i u ) ale : (x)e( u ale ) (u ) y i (u ) y io o(u yu ) 9 = ;. 3

17 We iet the above idetity back to the exeio of v (x) ad obtai that v (x) = ex (u )o(yu )O(y u )!(u ) u w y u u 4 o ex u u 4 E( 4 ale ) u H,x (u ), u (u ) y; u 8 h < E (x) 6 u ( u ) 3 (x) 4 ex u 4 ( i9 u ) ale = : (x)e( u ale ) ;, 4 o whee H,y (x, ; u), e x E h(y)(x ) (y) i u (x ) ale. Poof of emma 8. We iet = u to the exeio of i (3) ad obtai that = (u )o(yu )O(y u y ) w 4 u 4 u u G u (u ) y; u G ( ; u ) E[ () 6 u ( ) 3 () 4 u 4 ( ) ale ]. ()E( ale ) E 4 ale 4 u Note that = lim G (,u ). The, = (u )o(yu )O(y u y ) w 4 u 4 u u E[ () 6 u ( ) 3 () 4 u 4 ( ) ale ] ()E( ale ) o(). u (u ) y = (u )o(yu )O(y u y ) w ale u u u E 4 ale 4 u o() u (u ) y. whee ale i give a i (9). q q Poof of emma 9. I thi cae that u (u ) y >", the maximum of v (x) i ot eceaily attaied at x =. Note that thi doe ot chage the calculatio vey much excet that the tem (x) ad (x) ih x, may ot be evaluated o the bouday x =, 3

18 but till i the egio [ x[ u = G u /,] u /,]. Theefoe, maximiig (9), we have that u log H,x ( (u ), (u ) y; u u O(u / ). (u ) y; u ) Theefoe, we oly eed to add a O(u / ) to the defiitio of i (3). Futhemoe, the tem i (3) i bouded by G u (u ) y, u G (,u ) ale " fo ome >. Futhemoe, o the et u we have that (u )o(yu )O(y u )=o(). Theefoe, we have the boud S(w, y, ) u w (w) u / " u ad futhe P max v (x) >b; x[ u / u ;,] u (u ) y "! = o()u e u /. D Numeical Examle I thi ectio, we eet oe umeical examle. We coide the di eetial equatio i [,.5], that i, =.5. The Gauia oce ha eo mea ad uit vaiace. The covaiace fuctio i C(t) =e t / ad thu (x) i ifiitely di eetiable. Futhemoe, we coide a cotat foce (x) = ad thu F (x) =x. We comute the tail obability P (max x[,] v (x) >b) via the aoximatio i the theoem, deoted by w(b), ad futhemoe we comute the obabilitie via imotace amlig, deoted by ŵ(b). Fo the Mote Calo etimato, we chooe the amle ie uch that the etimated tadad deviatio of the etimato i at the mot % of ŵ(b). Figue?? how the atio betwee w(b)/ŵ(b) a a fuctio of log(b). The atio tabilie to oe a b become lage, but the covegece i quite low a the mallet obability i Figue?? i o the ode of 9. We futhe coide a ocotat foce tem (x) = max( (x.5), ) i the iteval [, 5]. The covaiace fuctio i C(t) =e.3t. The coeodig lot of w(b)/ŵ(b) veu log(b) i give by Figue??. The emiical ate of covegece of the o-cotat cae i much lowe tha that of the cotat cae. 3

19 w.tilde/w.hat log(b) Figue : The atio w(b)/ŵ(b) veu log(b) fo (x) = w.hat/w.tilde log(b) Figue 3: The atio ŵ(b)/ w(b) veu log(b) fo o-cotat (x). 33