A^VÇÚO 1 33 ò 1 1 Ï 217 c 2 Chinese Journal of Applied Probability and Statistics Feb., 217, Vol. 33, No. 1, pp. 49-57 doi: 1.3969/j.issn.11-4268.217.1.4 'uùk$ä ¼4 ½n[7,Âñ Ý oÿ 4[?>f EŒÆêÆ OŽ ÆÆ,?, 5414 Á : ^Œ ïäùk$ä ÛÜ ¼4 ½n, y²3[7,âñ ÂeÙK$ÄOþ'ur, p-nýûü ¼4 Âñ Ý. ' c: ÙK$Ä; [7,Âñ; Hölder ê; NÝ ã aò: O212.4 = Ú^ ª: Li Y H, Liu Y H. The rate of quasi sure convergence of a functional itation theorem for a Brownian motion [J]. Chinese J. Appl. Probab. Statist., 217, 331: 49 57. in Chinese 1. Ú ó [7, Û Å Û. glyoshida [1] 'ur, p-nýœ Ø, Æö m '5ÙK$Ä3Hölder êe[7, ¼4 nø. BaldiÚRoynette [2] ^Œ óäùk$ä3hölder êeâñ ÝJ, ChenÚBalakrishnan [3] ÙK$Ä'ur, p-ný ¼ éêæj, LiuÚRen [4] \rùk$ ÄOþ3Hölder êe'ur, p-nýœ J, 3Hölder ê r, p-nýeù K$Ä ¼ëYØ. ^Œ, ÙK$ÄOþ3[7,Âñ Âe'ur, p-nýûü ¼4 Âñ Ý. IJ;Wiener mb, H, µ, ér, p 1, D r,p Wiener ¼Sobolev m, = D r,p = 1 L r/2 L p, F r,p = 1 L r/2 F p, F L p, Ù L p P B, µþ Š¼êL p m, L B, H, µþornstein-uhlenbeckžf. é r, p > 1, r, p-ný½âxe: ém8o B, O = inf F p r,p; F D r,p, F 1, µ A??3 O þ, é? 8ÜA B, A = inf O; A O B, O m8. I[g, ÆÄ7 81OÒ: 7116217Ú2ÜŽg, ÆÄ7 81OÒ: 214GXNSFAA1181]Ï. 215c54FÂ, 215c1231FÂ?Uv.
5 A^VÇÚO 1 33 ò C d l[, 1]R d ëy¼ê m, Dƒþ. ê f = sup t1 ft, Ù L«îAp ê. P Ca d = f C d ; f = a, t H d = f C d ; ft = fsds, f 2 H = d H d ½ÂXeSÈHilbert m: ρ 1, ρ 2 H d = 1 1 ρ 1 s, ρ 2 sds. µ C d þwienerÿý, C d, H d, µ ²;Wiener m. Banach m KC α,, H d, µ C α = C α, = f C d ; f = f C α ; δ ft 2 dt <, ft fs sup s,t [,1] t s α <, s t sup s,t [,1] < t s <δ ft fs t s α =, é < α < 1/2, Äü ²;Wiener më z[2]½n2.4, ω C α, IOÙK$Ä. ½ÂI : B [, ] : ef H d, If = f 2 H d /2; ÄKIf =. PF = f H d ; If 1. BaldiÚRoynette [2] y²3k = kα,? Ú, é? f F k w f ε ε2/1 2α ln P τε ε 1/1 2α Ún 1 [3; ½n2.1] ε ε2/1 2α ln P w ε = k, 1 2. ýø = If kτ 2/2α 1. 2 S ε ε> C α, þ V 5Žf, é? ε > 9A C α, kµsε 1 A = µε 1/2 A. Kér, p [, 1,, k inf If ε ln Sε 1 A ε ln Sε 1 A inf If. ε ε f A f A Ún 2 [6; Ún2.1] k N, q 1, q 2 1, v1/p = 1/q 1 + 1/q 2. K3~ êc = ck, p, q 1, q 2 >, é? < a i < b i <, δ, 19F i D k,kq 1 k n C k,p a i < F 1/p i z < b i i=1
1 1 Ï oÿ, 4[ : 'uùk$ä ¼4 ½n[7,Âñ Ý 51 cn k δ k kµ n 1/q2, 1 + max F i k,kq1 a i δ < F i z < b i + δ 1in Ù F i F i [ëy?. Ún 3 k, p, q 1, q 2 XÚn2 ½Â. é? ε > 9f F, F ε i wti + h i wt i w = ε f, h i >, t i, i = 1, 2,..., n, hi K3 ~êc = ck, p, q 1, f >, é? δ, 1], ε, 1], k n 1/p n 1/q2. C k,p z : a i < F ε i z < b i cn k δ 2k2 k µ z : a i δ < F ε i z < b i + δ i=1 i=1 i=1 y²: ^Ún2, aq z[6] Ún2.2 y. Ún 4 f F, α k > X2ª ½Â, Két, τ > k wt + h wt f ε ε2/1 2α ln ετ h ε 1/1 2α = ε 2/1 2α wt + h wt f ln µ ετ ε h ε 1/1 2α = If kτ 2/2α 1. y²: ÏNÝäk5Ÿ µ, Iy²eª=Œ: wt + h wt f ε ε2/1 2α ln ετ h ε 1/1 2α ε 2/1 2α wt + h wt f ln µ ετ ε h ε 1/1 2α. q 1, q 2, p, kxún2 ½Â, k = [r] + 1, é? 1 > δ >, c >, ŠâÚn3k wt + h wt h f 1/p ετ ε 1/1 2α c ε 1/1 2α+1 δ 2k2 k ε 1/1 2α µ 2 ^2ª wt + h wt h 1/q2, f ε 1/1 2α+1 τ + δ wt + h wt f ε ε2/1 2α ln ετ h ε 1/1 2α p ε 2/1 2α f ln µ w ετ + δ q 2 ε ε 1/1 2α = p q 2 If + kτ + δ 2/2α 1, -þª δ, q 2 p, Ún4¼y.
52 A^VÇÚO 1 33 ò 3. ÙK$ÄOþ ¼ÛÜ[7,Âñ Ý x u lr + R + ünø~¼ê, Šâ z[8], bx u v < x u u u/x u ü NØ~. Ú\PÒρ u = lnux 1 u ln u, ét [, u x u ]P M t,u s = wt + x us wt x 1/2 u kρ u α 1/2, s [, 1], Ù kx2 ½Â. Lˆ Bå, Ú\PÒLLx = ln ln x. Ì J ½n8, TJy²dÚn5!Ún69Ún7. Ún 5 inf M t,us 1, - q.s. y²: é1 < θ < 1 ε 2, = θ n. À η > δ = 1/θ 1/2 1 ε 2/1 2α η > 1. gu = x u ρ u 2/1 2α, h n = [ /g ], t i = ig, i =, 1, 2,..., h n. Kk min M ti, s <ih n sup sup x 1/2 kρ un+1 1/2 α ws + t i + x un+1 wt i + x u ih n sg x 1/2 t [, x un+1 ] + inf kρ un+1 1/2 α wt + x un+1 wt = I 1 + I 2. 3 é? < ε < 1, dún4, n Œžk min M ti, s 1 ε ih n wt i + x un+1 wt i θ 1/2 1 ε ρ un+1 x u 1/2 ρ un+1 1 α ih n ÏdŠâBorel-CantelliÚn,, Ú\PÒ V j, n = 1 + h n exp ρ un+1 δ + g x un+1 δ, g ln n min M ti, s 1, - q.s. 4 ih n sup wt g + t i + jg + g wt i + jg + g, T 1
1 1 Ï oÿ, 4[ : 'uùk$ä ¼4 ½n[7,Âñ Ý 53 Ké? ε >, k I 1 > ε = x 1/2 kρ un+1 1/2 α [x un /g] j= [x un /g] j= h n i= h n i= sup sup ih n T 1 x 1/2 kρ un+1 1/2 α x α g α V j, n > ε k 1/2 α θ α+3/2 ρ un+1 1/2+α g V j, n > ε, wt g + t i + x un wt i + x un > ε éuf = f C α, ; sup f1/2t + 1/2 f1/2 ε, du t1 µ k 1/2 α θ α+3/2 ρ un+1 1/2+α g V j, n > ε inf f F If ε 2 /32, dún1, n Œžk k 1/2 α θ α+3/2 ρ un+1 1/2+α g V j, n > ε Än ž, ρ un+1, lk 2 k 1/2 α θ α+3/2 = µ ρ un+1 1/2+α w F, x un+1 ln ε2ρ un+1 2α/128θα+3/2. I 1 > ε n n x un+1 g + g x un+1 ε2ρ un+1 2α/128θα+3/2 <, g ln 2g ^Borel-CantelliÚn éü3 5ª? Ú, éuu [, k inf M t,us k 1/2 α ρ 1 α u I 1 =, - q.s. 5 n I 2 1, - q.s. 6 n inf x ρ un+1 1/2 wt + x un+1 wt t [, x un+1 ] 1 θ 21 α I 2, 7 3þª -θ 1, 2Ü6!7ªyÚn5.
54 A^VÇÚO 1 33 ò Ún 6 e lnu/x ullu 1 <, Kk inf M t,us 1, - q.s. y²: PA = x u/u, w,a 1. e ü«œ¹?ø. i ea < 1. ½ÂS, n 1Xe: u 1 = 1, = + x un+1, P w = wt n + x un wt n, t n = x un, léun 1, k x un ρ un 1/2 w > εƒpõá. é? ε >, λ = [x] + 1, dú n3k 1/p z M tn,un s 1 + ε z w = k 1 2α/2 ρ un x un 1/2 1 + ερ α 1 cz λ k 1 2α/2 ερ α 1 u z 2λ 2 λ? ÚŠâ2ª, n Œžk z w 1 µ 1/p < k 1 2α/2 ρ un x un 1/2 1 + ερ α 1 w µ < 1 + εk 1 2α/2 ρ un x un 1/2 ρ α 1 xun δ, ln Ù é, δ >, δ = 1 + ε 2/2α 1 + δ < 1. k z M tn,un s 1 + ε cz λ ρ 1 α 2λ2+λ u z exp εk 1 2α/2 1 q 2 éuþª?ê, Œ±y²3, π >, 1/p z xun δ > π ln u z 1 δ, ln z xun δ, ln 1/q2, qï lnu/x ullu 1 <, 3, N > u/x u ln u N. θ > 2/1 δ, u = e ln z θ, Kz vœ, z z ž, kln u z ln z θ, kln u z 1 δ > ln z 2. Ón, éc 1 >, z Œž, kc 1 z ln u z, k z 1/p M tn,un s 1 + ε cz λ LLu z 1 α2λ2 +λ exp π ln u z 1 δ q 2
1 1 Ï oÿ, 4[ : 'uùk$ä ¼4 ½n[7,Âñ Ý 55 c 2 z λ ln z 1 α2λ2 +λ exp π ln 2 z, z, q 2 Ù c 2 = c 2 λ, p, q 1, k, α.?k z=1 n=z M tn,un s 1 + ε =, dún6ø. ii ea = 1, Kx u = u, džy²ë z[3] ½n3.2. Ún6¼y. Ún 7 e lnu/x ullu 1 =, Kk inf M t,us 1, - q.s. y²: ŠâÚn7^ Œ3 S, n 1, v /x un = n p, p > 1. w, ün4o, n ž. t i = ix un+1, i =, 1, 2,..., v n = [ un ] 1, hn = ln/x un = ln np x un+1 LL LL, K = expn p /hn, n, hn. é? θ >, n kn θ ln 1, 1 = expn + 1 p hn+1 1 n p hn 1 expn p hn 1 1 1. ÄÀ β > β = 1 + ε 2/2α 1 + β < 1. -λ = [x] + 1, P Ω i = x un+1 ρ un+1 1/2 wt i + x un+1 wt i, KŠâÚn392ª, n Œžk inf M t, s 1 + 2ε t [, x un+1 ] min Ω i k1 2α/2 1 + 2ε iv n ρ un+1 1 α 1 + v n pλ ρ 1 α 1 + v n pλ ρ 1 α 1 + v n pλ ρ 1 α p2λ2+λ µ εk 1 2α/2 min Ω i k1 2α/2 1 + ε p/q2 iv n ρ un+1 1 α p2λ2+λ εk 1 2α/2 1 µ w < k1 2α/2 1 + ε 1+vn/q2 ρ un+1 1/2 α εk 1 2α/2 p2λ2+λ cn pλp lnn + 1 p1 α2λ2 +λ exp x un+1 1 ln β 1+v n/q 2 p q 2 [ un ] r β un+1 x un+1 ln,
56 A^VÇÚO 1 33 ò Ù ~êc >. éþªïlà p Œ cn pλp lnn + 1 p2λ2 +λ1 α exp p [ un ] x β un+1 <, q 2 x un+1 ln 2 ^Borel-CantelliÚnly3 4OS, n ž, éu [,, k n inf M t, s 1, - q.s. 8 t [, x un+1 ] inf M x un+1 ρ un+1 1/2 t,us inf t [, x un+1 ] x un ρ un 1/2 M t,un+1 s, Ä ÏdŠâ8ª? x un+1 ρ un+1 x un ρ un 1, n, inf M t,us 1, - q.s. ld/yún7. ½n 8 e lnu/x ullu 1 <, Kk e lnu/x ullu 1 =, Kk inf M t,us = 1, - q.s. 9 inf M t,us = 1, - q.s. 1 y²: ŠâÚn5!Ún6Œ9ª á, 2ŠâÚn5!Ún7Œ1ª á. ë z [1] Yoshida N. A large deviation principle for r, p-capacities on the Wiener space [J]. Probab. Theory Relat. Fields, 1993, 944: 473 488. [2] Baldi P, Roynette B. Some exact equivalents for the Brownian motion in Hölder norm [J]. Probab. Theory Relat. Fields, 1992, 934: 457 484. [3] Chen X, Balakrishnan N. Extensions of functional LIL w.r.t. r, p-capacities on Wiener space [J]. Statist. Probab. Lett., 27, 774: 468 473. [4] Liu J C, Ren J G. A functional modulus of continuity for Brownian motion [J]. Bull. Sci. Math., 27, 1311: 6 71.
1 1 Ï oÿ, 4[ : 'uùk$ä ¼4 ½n[7,Âñ Ý 57 [5] Wang W S. A generalization of functional law of the iterated logarithm for r, p-capacities on the Wiener space [J]. Stochastic Process. Appl., 21, 961: 1 16. [6] Liu Y H, Li L Q. The rate of quasi sure convergence in the functional it theorem for increments of a Brownian motion [J]. J. Math. Anal. Appl., 29, 3561: 21 29. [7] Baldi P, Arous G B, Kerkyacharian G. Large deviations and the Strassen theorem in Hölder norm [J]. Stochastic Process. Appl., 1992, 421: 171 18. [8] Csörgö M, Révész P. How big are the increments of a Wiener process? [J]. Ann. Probab., 1979, 74: 731 737. [9] Deuschel J D, Stroock D W. Large Deviations [M]. Boston: Academic Press, 1989. [1] Üá#. ÙK$Ä3r, p-ný Âee4 5Ÿ [J]. êææ, 1996, 394: 543 555. The Rate of Quasi Sure Convergence of a Functional Limitation Theorem for a Brownian Motion LI YuHui LIU YongHong School of Mathematics and Computing Science, Guilin University of Electric and Technology, Guilin, 5414, China Abstract: In this paper the local functional it theorem for increments of a Brownian motion is derived with large and small deviations, and the local functional convergence rate for increments of Brownian motion in Hölder norm with respect to r, p-capacity is estimated. Keywords: Brownian motion; quasi sure convergence; Hölder norm; capacity 21 Mathematics Subject Classification: 6F15; 6F17