Exponential inequalities and the law of the iterated logarithm in the unbounded forecasting game

Transcription

1 Ann Inst Stat Math (01 64: DOI /s Exponntial inqualitis and th law of th itratd logarithm in th unboundd forcasting gam Shin-ichiro Takazawa Rcivd: 14 Dcmbr 009 / Rvisd: 5 Octobr 010 / Publishd onlin: 30 Dcmbr 010 Th Institut of Statistical Mathmatics, Tokyo 010 Abstract W study th law of th itratd logarithm in th framwork of gamthortic probability of Shafr and Vovk W invstigat hdgs undr which a gam-thortic vrsion of th uppr bound of th law of th itratd logarithm holds without any condition on Rality s movs in th unboundd forcasting gam W prov that in th unboundd forcasting gam with an xponntial hdg, Skptic can forc th uppr bound of th law of th itratd logarithm without conditions on Rality s movs W giv two xampls such a hdg For proving ths rsults w driv xponntial inqualitis in th gam-thortic framwork which may b of indpndnt intrst Finally, w giv rlatd rsults for masur-thortic probability which improv th rsults of Liu and Watbld (Stochastic Procsss and thir Applications 119: , 009 Kywords Exponntial inquality Gam-thortic probability Law of th itratd logarithm 1 Introduction In this papr, w invstigat th law of th itratd logarithm in th framwork of gam-thortic probability of Shafr and Vovk (001 W considr th following protocol S Takazawa (B Dpartmnt of Mathmatics, Graduat School of Scinc, Kob Univrsity, 1-1 Rokkodai, Nada-ku, Kob , Japan -mail: takazawa@mathkob-uacjp 13

2 616 S Takazawa Th Unboundd Forcasting Gam with a Hdg Protocol: K 0 := 1 FOR n = 1,,: Forcastr announcs v n > 0 Skptic announcs M n R and V n 0 Rality announcs x n R K n := K n 1 + M n x n + V n (h(x n v n END FOR In this gam, in ordr to prvnt Skptic from bcoming infinitly rich, Rality is forcd to bhav probabilistically In Chaptr 4 of Shafr and Vovk (001, a gamthortic vrsion of th strong law of larg numbrs is provd undr th varianc hdg (quadratic hdg h(x = x Namly, in th unboundd forcasting gam with th quadratic hdg h(x = x, Skptic has a stratgy that dos not risk bankruptcy and allows him to bcom infinitly rich if (x x n /n 0 Kumon t al (007 stablishd a gam-thortic vrsion of th strong law of larg numbrs undr othr typs of nonngativ symmtric hdgs h(x such that n>c 1/h(n < for som c 0 In addition, Takazawa (009 proposd an xponntial inquality in th unboundd forcasting gam with th hdg h(x = x As rgards th convrgnc rat of th strong law of larg numbrs in gams with boundd variabls, thr xist various rsarchs (s Horikoshi and Takmura (008, Kumon and Takmura (008, Kumon t al (008 InKumon and Takmura (008, it is provd that in th boundd forcasting gam a simpl singl stratgy that is basd only on th past avrag of Rality s movs wakly forcs th strong law of larg numbrs with th convrgnc rat of O( log n/n InKumon t al (008, it is provd that th Baysian stratgy of Skptic wakly forcs th strong law of larg numbrs with th convrgnc rat of O( log n/n in coin-tossing gams Horikoshi and Takmura (008 considrd th lowr bound on th convrgnc rat of th strong law of larg numbrs in fair-coin gam Morovr, Vovk (007 prsntd a gam-thortic vrsion of Azuma Hoffding s inquality in th boundd forcasting gam In Chaptr 5 of Shafr and Vovk (001, a gam-thortic vrsion of th law of th itratd logarithm is provd undr th varianc hdg h(x = x Thorm 1 (Thorm 5 of Shafr and Vovk (001 In th unboundd forcasting gam with th quadratic hdg h(x = x, Skptic can forc A n and x n =o ( A n log log A n whr S n = n i=1 x i and A n = n i=1 v i lim sup S n An log log A n 1, This rsult corrsponds to th uppr bound of Kolmogorov s law of th itratd logarithm (s g Ptrov (1995 and Stout s martingal vrsion of th uppr bound of th law of th itratd logarithm (s Stout (1970 In Kolmogorov s and Stout s law 13

3 Exponntial inqualitis and LIL for unboundd forcasting gam 617 of th itratd logarithm, th uppr bound and th lowr bound hold undr th sam conditions On th othr hand, in th unboundd forcasting gam with th quadratic hdg h(x = x, th lowr bound of th law of th itratd logarithm dos not hold (s Proposition 51 of Shafr and Vovk (001 In ordr to hav th lowr bound, w must mak th unboundd forcasting protocol mor favorabl to Skptic, that is, w must slightly rstrict Rality s frdom of action and slightly incras Skptic s frdom of action, as follows: Th Prdictably Unboundd Forcasting Gam Protocol: K 0 := 1 FOR n = 1,,: Forcastr announcs v n > 0 and c n 0 Skptic announcs M n R and V n R Rality announcs x n c n K n := K n 1 + M n x n + V n (xn v n END FOR Thorm (Thorm 51 of Shafr and Vovk (001 In th prdictably unboundd forcasting gam, Skptic can forc ( A n A n and c n = o log log A n lim sup S n An log log A n = 1, whr S n = n i=1 x i and A n = n i=1 v i This is on of th diffrncs btwn masur-thortic probability and gamthortic probability In th gam-thortic framwork, it is natural to considr th uppr bound and lowr bound of th law of th itratd logarithm undr diffrnt conditions, bcaus Skptic can forc th uppr bound undr wakr conditions than th lowr bound In this papr, w invstigat hdgs undr which a gam-thortic vrsion of th uppr bound of th law of th itratd logarithm holds without any condition on Rality s movs in th unboundd forcasting gam In th viw of gam-thortic probability, th conditional statmnt of Thorm 1 is not dsirabl bcaus Rality s movs and Forcastr s movs cannot ncssarily b limitd Thrfor, it is of intrst to stablish protocols undr which th law of th itratd logarithm holds undr conditions favorabl to Rality In particular, if w considr only th uppr bound of th law of th itratd logarithm, it is natural to drop conditions on Rality s movs Th following thorm is our rsult in this dirction Thorm 3 Lt S n = n i=1 x i and A n = n i=1 v i Thn, in th unboundd forcasting gam with th hdg h(x = x 1, for any t ( 1, 1, th procss xp (ts n t 1 t An is a gam-thortic suprmartingal Furthrmor, st B n = 1 n i=1 v i ( = 1 A n 13

4 618 S Takazawa Thn, Skptic can forc lim B n < lim sup S n <, lim B S n n = lim sup 1 Bn log log B n Th notion of a gam-thortic suprmartingal will b dfind in Sct Th first part of Thorm 3 says that in th unboundd forcasting gam with th hdg h(x = x 1, for any t ( 1, 1, Skptic can guarant K n = xp (ts n t 1 t An, if h is allowd to discard part of his capital at ach stp If w translat Thorm 3 in masur-thortic trms, w hav th following rsult Thorm 4 Lt (X i b a martingal diffrncs squnc and S n = n i=1 X i St A n = n i=1 E [ X i 1 F i 1 ] For all t ( 1, 1, lt V n = xp (ts n t 1 t An Thn V n is a positiv suprmartingal with E[V n ] 1 Furthrmor, st B n = 1 A n If B n < as for all n and lim B n = as, thn lim sup S n Bn log log B n 1 as At th sam tim, by th first part of Thorm 4, w hav an xponntial inquality in th masur-thortic framwork Thorm 5 Lt (X i b a martingal diffrncs squnc and S n = n i=1 X i St A n = n i=1 E [ X i 1 F i 1 ] Thn, for all a, b > 0, ( } P( S n a, A n b xp a + 1 b 1 b (1 In gam-thortic probability, w can dfin uppr probability (s Sct Thn, from Thorm 3, w hav an xponntial inquality in th gam-thortic framwork Thorm 6 In th unboundd forcasting gam with th hdg h(x = x 1,ifall v n ar givn in advanc, thn for all a > 0, ( P( S n a xp a + 1 A n } 1 A n, ( whr S n = n i=1 x i and A n = n i=1 v i 13

5 Exponntial inqualitis and LIL for unboundd forcasting gam 619 Not that although Thorm 6 includs th assumption that all v n ar givn in advanc, Thorm 3 dos not includ th assumption W mntion that our gam-thortic rsult ( is simpl compard with a corrsponding masur-thortic rsult (1 This rprsnts ffctivnss of gam-thortic probability Furthrmor, in th cas whr h(x = xp(x / 1, w hav similar rsults to th cas whr h(x = x 1 W mntion that th lowr bound of th law of th itratd logarithm simply dos not hold in th gnrality of th unboundd forcasting gam with a hdg h(x = x 1 and h(x = xp(x / 1 This is bcaus th argumnt in Shafr and Vovk (001, Proposition 51, still applis Thrfor, this papr concntrats on th uppr bound of th law of th itratd logarithm Th organization of th rst of this papr is as follows: In Sct, wgivsom notations on gam-thortic probability In Sct 3, w propos an xponntial inquality and prov th uppr bound of th law of th itratd logarithm in th unboundd forcasting gam with th hdg h(x = x 1 In Sct 4, w propos an xponntial inquality and prov th uppr bound of th law of th itratd logarithm in th unboundd forcasting gam with th hdg h(x = xp(x / 1 Finally, in Sct 5, w provid th rlatd rsults for masur-thortic probability Notation In this sction, w giv som notations on gam-thortic probability Th Unboundd Forcasting Gam with a Hdg Protocol: K 0 := 1 FOR n = 1,,: Forcastr announcs v n > 0 Skptic announcs M n R and V n 0 Rality announcs x n R K n := K n 1 + M n x n + V n (h(x n v n END FOR Skptic starts with th initial capital of K 0 = 1 In ach round n, Forcastr announcs v n > 0 first Nxt Skptic announcs M n R and V n 0 and Rality dcids x n R aftr sing Skptic s mov M n and V n In this protocol, w considr a hdg h(x n It is nonngativ and has a finit pric 0 <v n < Skptic is only allowd to buy arbitrary amount of this hdg, that is, V n is rstrictd to b nonngativ (V n 0 Th st of all infinit squncs v 1 x 1 v x of Forcastr s and Rality s movs is calld sampl spac ω = v 1 x 1 v x dnots an infinit squnc of Forcastr s and Rality s movs and ω n = v 1 x 1 v x v n x n dnots a squnc of Forcastr s and Rality s movs up to round n An vnt E is a subst of W dnot by K n Skptic s capital at th nd of round n For a stratgy P of Skptic, Kn P (ω dnots th capital procss of P W say that P satisfis th collatral duty if its capital procss is always nonngativ, that is, if 13

6 60 S Takazawa K P n (ω 0, ω, n 0 W also say that P is prudnt if it satisfis th collatral duty Whn P is prudnt, th capital procss K P is calld a (gam-thortic nonngativ martingal A procss S is a ral-valud function of v 1 x 1 v n x n in th unboundd forcasting gam W say that a procss S is a gam-thortic suprmartingal if thr is a stratgy for Skptic that guarants S(v 1 x 1 v n x n S(v 1 x 1 v n 1 x n 1 K n K n 1, for all n and for all v 1 x 1 v x ; in othr words, a gam-thortic suprmartingal is a possibl capital procss for Skptic who is allowd to discard part of his capital at ach stp W say that Skptic can wakly forc an vnt E if thr xists a prudnt stratgy P of Skptic such that lim sup Kn P (ω =, ω E Similarly, w say that Skptic can forc an vnt E if thr xists a prudnt stratgy P of Skptic such that lim KP n (ω =, ω E Shafr and Vovk (001 drivd th following two lmmas Lmma 1 (Lmma 31 of Shafr and Vovk (001 If Skptic can wakly forc E, thn h can forc E Lmma (Lmma 3 of Shafr and Vovk (001 If Skptic can wakly forc ach of a squnc E 1, E,of vnts, thn h can wakly forc k=1 E k Following Shafr and Vovk (001, Takuchi (004, and Takmura t al (009, for an vnt E, w dfin uppr probability P(E as P(E = inf α 0 Thr xists a prudnt stratgy P of Skptic such that sup Kn } P(ω 1/α for all ω E n 1 Not that 0 P(E 1, bcaus Skptic can choos M n = 0 and V n = 0 for all n In addition, if Skptic can forc E, thn P(E C = 0 3 An xponntial inquality and LIL in th unboundd forcasting gam with th hdg h(x = x 1 In this sction, w considr th unboundd forcasting gam with th hdg h(x = x 1 13

7 Exponntial inqualitis and LIL for unboundd forcasting gam 61 Th Unboundd Forcasting Gam I Protocol: K 0 := 1 FOR n = 1,,: Forcastr announcs v n > 0 Skptic announcs M n R and V n 0 Rality announcs x n R K n := K n 1 + M n x n + V n ( xn 1 v n END FOR W start with th following thorm This thorm is inspird by Lmma 6 in Liu and Watbld (009 Thorm 7 In th unboundd forcasting gam I, for any t ( 1, 1, th procss n i=1 is a gam-thortic suprmartingal xp (tx i t 1 t vi Proof W considr th following stratgy P: M n = t xp V n = 1 v n 1 xp 1 t vn K n 1, 1 t vn } K n 1 Undr this stratgy, Skptic s capital procss Kn P, starting with th initial capital of K 0 = 1, is writtn as K P n = n i=1 [ xp 1 t vi + tx i xp + 1 v i ( x i 1 1 xp 1 t vi 1 t vi } ] In ordr to prov th thorm, it suffics to show that for any t ( 1, 1, xp (tx t 1 t v xp + 1 v 1 t v ( x 1 1 xp + tx xp 1 t v 1 t v } 13

8 6 S Takazawa For x 0 and k 0, lt f k (x = x 1 1 xk k! Thn w hav f k (0 = 0 for all k, and for k 1, Sinc w hav f k (x = x 1 xk 1 (k 1! = f k 1(x + 1 f 1 (x = x 1 1, f (x f 1(1 + 1 = 0 It follows that f (x 0 for any x 0 Hnc, for all k, w hav Thrfor for 1 < t < 1, tx = 1 + tx tx + f k (x f k (0 = 0 t k x k k= k! ( t k x 1 1 k= = 1 + tx + t 1 t Sinc y y 1 y for y 0, w hav 13 xp (tx t 1 t v xp + t xp + 1 v 1 t ( x 1 1 (3 + tx xp 1 t v ( x 1 1 xp 1 t v + tx xp ( x 1 1 xp 1 t v 1 t v 1 t v } 1 t v

9 Exponntial inqualitis and LIL for unboundd forcasting gam 63 Thrfor, th procss n i=1 xp (tx i t 1 t vi is a gam-thortic suprmartingal From Thorm 7, w hav th following rsult similar to Thorm 41 of Liu and Watbld (009 Thorm 8 Lt S n = n i=1 x i and A n = n i=1 v i In th unboundd forcasting gam with th hdg h(x = x 1, ifallv n ar givn in advanc, thn for all a > 0, ( P( S n a xp a + 1 A n } 1 A n Proof From Thorm 7,for0 t < 1, Skptic s capital K n is boundd as K n n i=1 xp on th vnt S n a }For0 t < 1, lt Thn f (t = 0 if and only if f (t = ta xp (tx i t 1 t vi, (ta t 1 t An t 1 t An t = 1 1 A n a + 1 A n Sinc ( f 1 1 A n ( = a + 1 A n 1 A n, a + 1 A n w hav ( K n xp a + 1 A n } 1 A n Thrfor w obtain P(S n a xp ( a + 1 A n } 1 A n 13

10 64 S Takazawa W can find th sam uppr-bound for P(S n a Hnc w hav ( P( S n a xp a + 1 A n } 1 A n Rmark 1 W mntion that Thorm 8 is strongr than Liu and Watbld s (43 in Thorm 41 This is bcaus Liu and Watbld s formula (43 can b writtn, in th sam notation, as ( P( S n > a xp a + An + n } A n + n Furthrmor, by an argumnt similar to Thorm 5 of Shafr and Vovk (001, w obtain th law of th itratd logarithm in th unboundd forcasting gam with th hdg h(x = x 1 Thorm 9 Lt S n = n i=1 x i and B n = 1 n i=1 v i Thn, in th unboundd forcasting gam with th hdg h(x = x 1, Skptic can forc lim B n < lim sup S n <, lim B S n n = lim sup 1 Bn log log B n Proof By Thorm 7, thr xists a stratgy for Skptic P(1/ such that It follows that K P(1/ n lim B n < and lim sup Thrfor Skptic can forc ( 1 xp S n 1 4 B n S n = lim sup Kn P(1/ = lim B n < lim sup S n < Nxt, w will show that Skptic has a prudnt stratgy P such that lim B n = and lim sup S n Bn log log B n > 1 lim KP n = (4 Lt E =ω lim B n = }and 13 E 0 = S n (1 + δ ω lim sup > } Bn log log B n 1 ε

11 Exponntial inqualitis and LIL for unboundd forcasting gam 65 In ordr to prov (4, it suffics to show that for any δ (0, 1 and for any ε (0, 1, thr xists a prudnt stratgy P of Skptic such that for all ω E E 0, lim sup Kn P (ω = Fix tmporarily a numbr ε (0, 1 For any θ (0,ε, w considr th following stratgy P (θ with initial capital 1: M (θ n = θ xp ( θ V (θ n = 1 v n 1 xp From Thorm 7, Skptic s capital K (θ n K (θ n 1 θ vn K (θ n 1, ( θ 1 θ vn satisfis ( xp θ S n ( xp θ S n θ (1 θ B n θ (1 ε B n } K (θ n 1 For any δ (0, 1, st } n k = min n B n (1 + δ k, k(n = log 1+δ B n Thn, for ach n, thr xists r(n [0, 1 such that k(n = log 1+δ B n r(nlt θ(k = (1 ε(1 + δ k log k Thn thr xists a positiv intgr N(δ, ε such that θ(k <εfor all k N(δ, ε For n k m < n k+1, (1 εbm 1 log log B m 1 + δ θ(k(m Thrfor for sufficintly larg m, 1 (1 ε log log B m θ(k(m 1 + δ B m ( log (1 ε(1 + δ r(m Bm 1 Bm log log(1 + δ (1 ε(1 + δ log log B m B m 13

12 66 S Takazawa Hnc w hav K (θ(k(m m xp (θ(k(ms m θ(k(m xp = xp ( S m (1 ε B m (1 ε log log B m (1 + δ log log B m B m 1 + δ ( 1 ε S m (1 + δ 1 Bm log log B m } (1 + δ log log B m, for sufficintly larg m St E k = n k+1 1 m=n k ω S m (1 + δ > } Bm log log B m 1 ε Dfin S m (1 + δ τ k = min n k m < n k+1 > }, Bm log log B m 1 ε ltting min =n k+1 1 W considr th following stratgy P [k] : M [k] n = V [k] n = θ(k xp ( θ(k (1 θ(k v n K (θ(k n 1, n τ k, 0, n >τ k, } 1 1 xp ( θ(k v n (1 θ(k v n K (θ(k n 1, n τ k, 0, n >τ k For sufficintly larg n, Skptic s capital K [k] n satisfis 13 K [k] n xp ( (1 + δ log log B nk xp ((1 + δ log log(1 + δ k = (k log(1 + δ 1+δ,

13 Exponntial inqualitis and LIL for unboundd forcasting gam 67 on E k Lt It follows that K 0 = k=n(δ,ε P(δ, ε = 1 K 0 K P(δ,ε n = 1 K 0 k=n(δ,ε k (1+δ, k=n(δ,ε k (1+δ P [k], k (1+δ K [k] n lim B S n (1 + δ n = and lim sup > Bn log log B n 1 ε lim sup Kn P(δ,ε = Thrfor, by Lmma 1 and Lmma, Skptic can forc lim B S n n = lim sup 1 Bn log log B n Th thorm is provd 4 An xponntial inquality and LIL in th unboundd forcasting gam with th hdg h(x = xp(x / 1 In this sction, w considr th unboundd forcasting gam with th hdg h(x = xp(x / 1 Th Unboundd Forcasting Gam II Protocol: K 0 := 1 FOR n = 1,,: Forcastr announcs v n > 0 Skptic announcs M n R and V n 0 Rality announcs x n R K n := K n 1 + M n x n + V n (xp(xn / 1 v n END FOR First w show th following thorm Thorm 10 In th unboundd forcasting gam II, for any t ( 1, 1, th procss n i=1 xp (tx i t 1 t v i 13

14 68 S Takazawa is a gam-thortic suprmartingal Proof W considr th following stratgy P: } M n = t xp 1 t v n K n 1, V n = 1vn 1 xp 1 t v n K n 1 Undr this stratgy, Skptic s capital procss Kn P, starting with th initial capital of K 0 = 1, is writtn as K P n = n i=1 [ xp + 1 v i xp 1 t v i ( x i + x i t xp 1 t v i 1} 1 xp 1 t v i } ] In ordr to prov th thorm, it suffics to show that for any t ( 1, 1, xp (tx t 1 t v xp 1 t v For x 0 and k, lt Thn, for all j 1, w hav + 1 v xp ( x + xt xp } 1 ( x g k (x = xp 1 xk k! 1 xp 1 t v 1 t v ( x g j xp (x = x x j } ( j 1! ( x ( 1 x j 1 } x xp 0, ( j 1! } and ( x g j+1 xp (x = x x j 1 } ( j! x ( x xp x j ( j! x j } ( j! x ( x xp 1 ( x j ( 1 x j 1 } 0 j! ( j 1! 13

15 Exponntial inqualitis and LIL for unboundd forcasting gam 69 Hnc, for all k, w hav Thrfor for 1 < t < 1, tx = 1 + tx tx + g k (x g k (0 = 0 t k x k k= k! = 1 + tx + t 1 t Sinc y y 1 y for y 0, w hav ( x t xp k k= ( x xp xp (tx t 1 t v xp 1 t v + t xp 1 t Thrfor, th procss xp + 1 v n i=1 } 1 } 1 + xt xp 1 t v ( x } 1 xp 1 t v 1 t v + xt xp 1 t v ( x } } 1 1 xp 1 t v xp xp (tx i t 1 t v i is a gam-thortic suprmartingal By th sam argumnts as Thorm 8 and Thorm 9, in th unboundd forcasting gam with th hdg h(x = xp(x / 1, w obtain an xponntial inquality and th law of th itratd logarithm, rspctivly Thorm 11 Lt S n = n i=1 x i and A n = n i=1 v i In th unboundd forcasting gam with th hdg h(x = xp(x / 1, ifallv n ar givn in advanc, thn for all a > 0, ( P( S n a xp a + An } A n 13

16 630 S Takazawa Thorm 1 Lt S n = n i=1 x i and C n = n i=1 v i Thn, in th unboundd forcasting gam with th hdg h(x = xp(x / 1, Skptic can forc lim C n < lim sup S n <, lim C S n n = lim sup 1 Cn log log C n 5 Th rlatd rsults for masur-thortic probability In this sction, w giv xponntial inqualitis and th uppr bound of th law of th itratd logarithm for a squnc of martingal diffrncs W obtain th following masur-thortic rsults from Thorms 7 9 Thorm 13 Lt (X i b a squnc of martingal diffrncs and S n = n i=1 X i St and B n = 1 A n (1 For all t ( 1, 1, lt A n = n E i=1 [ X i 1 F i 1 ], V n = xp (ts n t 1 t An Thn V n is a positiv suprmartingal with E[V n ] 1 ( For all a, b > 0, ( } P( S n a, A n b xp a + 1 b 1 b (3 If B n < as for all n and lim B n = as, thn lim sup S n Bn log log B n 1 as Thorm 13 can b drivd from Thorms 7 9 along th lins of Corollaris in Shafr and Vovk (001 This is bcaus stratgis for Skptic constructd in th proof of Thorms 7 9 ar masurabl Of cours, w can provid th masur-thortic proof of Thorm 13 indpndnt of th gam-thortic probability W can driv an xponntial inquality for a squnc of martingal diffrncs by an argumnt similar to Thorm 1 of Brcu and Touati (008 Also, w can prov th uppr bound of th law of th itratd logarithm for a squnc of martingal diffrncs by an argumnt similar to Thorm 11 of Stout (1973, or by using Corollary 4 of d la Pña t al (004 13

17 Exponntial inqualitis and LIL for unboundd forcasting gam 631 Rmark In Corollary 5 of Liu and Watbld (009, it is provd that if (X i is a squnc of martingal diffrncs such that for som constants λ i > 0, E[ X i F i 1 ] λ i as, (5 thn w hav lim sup S n n log n L as, (6 whr S n = n i=1 X i and L = lim sup (λ 1 + +λ n /n On th othr hand, Thorm 13 drivs th uppr bound of th law of th itratd logarithm for a squnc of martingal diffrncs without th bounddnss condition (5 for th conditional xponntial momnts In addition, this rsult is sharpr than (6 Similarly, by Thorms 10 1, w hav th following thorm Thorm 14 Lt (X i b a squnc of martingal diffrncs and S n = n i=1 X i St [ ( ] n X A n = E xp i 1 F i 1, and C n = A n i=1 (1 For all t ( 1, 1, lt V n = xp (ts n t 1 t A n Thn V n is a positiv suprmartingal with E[V n ] 1 ( For all a, b > 0, ( a } P( S n a, A n b xp + b b (3 If C n < as for all n and lim C n = as, thn lim sup S n Cn log log C n 1 as Rmark 3 In Thorm 4 of Liu and Watbld (009, it is provd that if (X i is a squnc of martingal diffrncs such that for som constants λ i > 0 and R > 0, E [ ] xp(rxi F i 1 λ i as, (7 13

18 63 S Takazawa thn for ach L (λ 1 + +λ n /n, thr xist a constant c > 0 dpnding only on L and R such that for all t R, E[ ts n ] xp(cnt, and for all a > 0, P( S n > a xp ( a, 4cn whr S n = n i=1 X i On th othr hand, Thorm 14 drivs an xponntial inquality and th uppr bound of th law of th itratd logarithm for a squnc of martingal diffrncs without th bounddnss condition (7 for th conditional xponntial momnts Acknowldgmnts Th author thanks anonymous rfrs for vry carful rading of th manuscript and for many valuabl commnts which improvd th papr Rfrncs Brcu, B, Touati, A (008 Exponntial inqualitis for slf-normalizd martingals with applications Th Annals of Applid Probability, 18, d la Pña, V H, Klass, M J, Lai, T L (004 Slf-normalizd procsss: xponntial inqualitis, momnts bounds and itratd logarithm laws Th Annals of Probability, 3, Horikoshi, Y, Takmura, A (008 Implications of contrarian and on-sidd stratgis for th fair-coin gam Stochastic Procsss and thir Applications, 118, Kumon, M, Takmura, A (008 On a simpl stratgy wakly forcing th strong law of larg numbrs in th boundd forcasting gam Annals of th Institut of Statistical Mathmatics, 60, Kumon, M, Takmura, A, Takuchi, K (007 Gam-thortic vrsions of strong law of larg numbrs for unboundd variabls Stochastics, 79, Kumon, M, Takmura, A, Takuchi, K (008 Capital procss and optimality proprtis of a Baysian Skptic in coin-tossing gams Stochastic Analysis and Applications, 6, Liu, Q, Watbld, F (009 Exponntial inqualitis for martingals and asymptotic proprtis of th fr nrgy of dirctd polymrs in random nvironmnt Stochastic Procsss and thir Applications, 119, Ptrov, V V (1995 Limit thorms of probability thory: squncs of indpndnt random variabls Oxford: Oxford Univrsity Prss Shafr, G, Vovk, V (001 Probability and financ: it s only a gam! Nw York: Wily Stout, W F (1970 A martingal analogu of Kolmogorov s law of th itratd logarithm Zitschrift für Wahrschinlichkitsthori und Vrwandt Gbit, 15, Stout, W F (1973 Maximal inqualitis and th law of th itratd logarithm Th Annals of Probability, 1, 3 38 Takazawa, S (009 An xponntial inquality and th convrgnc rat of th strong law of larg numbrs in th unboundd forcasting gam (Submittd Takmura, A, Vovk, V, Shafr, G (009 Th gnrality of th zro-on laws Annals of th Institut of Statistical Mathmatics, to appar doi:101007/s Takuchi, K (004 Mathmatics of btting and financial nginring (in Japans Tokyo: Sainsusha Vovk, V (007 Hoffding s inquality in gam-thortic probability arxiv: