THE LAW OF THE ITERATED LOGARITHM FOR TAIL SUMS SANTOSH GHIMIRE. M.Sc., Tribhuvan University, Nepal, 2001 M.S., Kansas State University, USA, 2008

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1 THE LAW OF THE ITERATED LOGARITHM FOR TAIL SUMS by SANTOSH GHIMIRE MSc, Tribhuvan University, Nepal, 00 MS, Kansas State University, USA, 008 AN ABSTRACT OF A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 0

2 Abstract The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it Sum of Rademacher functions is a nicely behaved dyadic martingale With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales We obtain both upper and lower bound in the case of martingales A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund

3 THE LAW OF THE ITERATED LOGARITHM FOR TAIL SUMS by SANTOSH GHIMIRE MSc, Tribhuvan University, Nepal, 00 MS, Kansas State University, USA, 008 A DISSERTATION submitted in partial fulfillment of the requirements for the degree DOCTOR OF PHILOSOPHY Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 0 Approved by: Major Professor Prof Charles N Moore

4 Copyright Santosh Ghimire 0

5 Abstract The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it Sum of Rademacher functions is a nicely behaved dyadic martingale With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales We obtain both upper and lower bound in the case of martingales A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund

6 Table of Contents Table of Contents List of Figures Acknowledgements Dedication vi vii viii ix Introduction Dyadic martingales Notation 3 Examples 4 4 Useful results 4 5 Origin of law of the iterated logarithm 9 6 Organization of the thesis 6 Law of the iterated logarithm 7 Martingales inequalities 7 Law of the iterated logarithm for dyadic martingales 5 3 The tail law of the iterated logarithm 8 3 The tail law of the iterated logarithm for sums of Rademacher functions 8 3 The tail law of the iterated logarithm for dyadic martingales Tail LIL for dyadic martingales is not true in general 50 4 Lower bound in the tail law of the iterated logarithm 60 4 Lower bound in the tail LIL for sums of Rademacher functions 60 4 The tail law of the iterated logarithm for independent random variables 7 43 Lower bound in the tail law of the iterated logarithm for dyadic martingales 89 5 The tail law of the iterated logarithm for lacunary series 03 5 Lower bound in the tail law of the iterated logarithm for lacunary series 03 Conclusion Bibliography vi

7 List of Figures Rademacher functions 4 3 Construction of Martingales 53 vii

8 Acknowledgments I would like to tha my advisor Professor Charles N Moore for his continuous support, invaluable advice, and time throughout my student career at K-state I consider myself very fortunate to have him as my advisor This project was not possible without his help Next, I would like to tha my committee members Professor Pietro Poggi-Corradini and Professor Marianne Korten They have been very friendly, supportive and helpful throughout my study at K-state I would like to tha you Professor Haiyan Wang, who is also my Statistics Master s major advisor for her help and continuous support My thas also goes to Professor William Hsu for serving as outside chair person of my committee I would like to tha the Department of Mathematics, KSU for providing me an opportunity to pursue my PhD degree I am thaful to my mother Jyanti Devi Ghimire for her continuous motivation and encouragement for my further study Her inspiration has made me come to this stage of my study My sincere thas goes to my wife Sarala Acharya who has been very supportive for my study Her continuous support and help have made me achieve this goal I would also tha her for patience and taking care of our daughter, Subi Ghimire and letting me focus on my work Finally, I would like to tha my brother Krishna Ghimire and entire Ghimire family for their support and motivation viii

9 Dedication I would like to dedicate this to: My father late Madhusudan Upadhayaya My mother Jyanti Devi Ghimire My daughter Subi Ghimire ix

10 Chapter Introduction This chapter begins with some useful definitions and notation which will be repeatedly used in later chapters We state some useful results and then discuss the origin of law of the iterated logarithm Dyadic martingales Before we define dyadic martingales, we discuss the meaning of the word martingale Originally martingale meant a strategy for betting in which you double your bet every time you lose Let us consider a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails The strategy is that the gambler doubles his bet every time he loses and continues the process, so that the first win would recover all previous losses plus win a profit equal to the original stake This process of betting can be represented by a sequence of functions which is an example of dyadic martingale Since a gambler with infinite wealth will, almost surely, eventually flip heads, the martingale betting strategy can be seen as a sure thing Of course, no gambler in fact possesses infinite wealth, and the exponential growth of the bets will eventually barupt those who choose to use the martingale strategy A dyadic interval of the unit cube [0, is of the form Q nj = [ j, j + for n, j Z n n We reserve the symbol Q n to denote a generic dyadic interval of length n Let F n denote

11 [ j the σ algebra generated by the dyadic intervals of the form, j + on [0, and let n n Ef n+ F n denote the conditional expectation of f n+ on F n which is defined as, Ef n+ F n x = Q n Q n f n+ ydy, x Q n Definition Dyadic martingales A dyadic martingale is a sequence of integrable functions f n } n=0, with f n : [0, R such that for every n, f n is F n measurable and Ef n+ F n = f n for all n 0 The sequence f n } n=0 is called a dyadic submartingale resp supermartingale if we replace = by resp in the expectation condition Here F 0 = [0,, φ}, F = [0,, φ, [0, /, [/, } and so on Hence f n is measurable with respect to F n means for all a R, the set x : f n x > a} belongs to F n Consequently, the function f n is constant on each of the n th generation dyadic intervals Q n The expectation condition tells us that the f n is the average of f n+ on Q n Moreover, the existence of the conditional expectation can be justified by Radon-Nikodym Theorem If we thi of f n } n=0 as the fortune of gambler at the instant n of a game, then the first condition simply says the trivial fact that the result of the game totally determines the state of fortune at any instant The second condition expresses that the game is fair in the sense that the expected fortune after any trial must be same as that of the fortune before the trial Notation For a dyadic martingale we have the following standard associated functions i Maximal function, f m = sup f k, f = sup f k k m k< ii Martingale difference sequence, d k }, where d k x = f k x f k x iii Martingale square function or Quadratic characteristics, n Snfx = S n fx = d kx k=

12 iv Martingale tail square function, S fx = Sfx = S n fx = S nfx = d kx k= k=n+ d kx We note that d 0 nxdx = 0 For this let Q nj be a dyadic interval of length Then we n have, 0 d n xdx = = n j= n j= d n xdx Q nj Using the fact that f n is constant on Q nj, we have 0 d n xdx = = = = 0 n j= n j= n j= [ Q nj [f n x f n x]dx f n xdx f n x Q nj Q nj [ ] Q nj f n xdx f n x Q nj Q nj Q nj [f n x Q nj f n x Q nj ] Also we have, f n x = n k= d kx + f 0 The martingale square function is a local version of variance and can also be understood as a discrete counterpart of the area function in Harmonic Analysis They play an important role in the study of asymptotic behavior of dyadic martingales We will see that the asymptotic behavior of a dyadic martingale is governed by the size of its quadratic variation From the definition, we note that for any x, y Q n, we have S nfx = S nfy But the martingale tail square function, S n fx may not be equal to S n fy ] 3

13 3 Examples Here we give some examples of dyadic martingales Example The functions r k } k= defined on [0, by r kx = sgnsin k πx where sgn is given by sgnx = for x 0 and sgnx = for x < 0 are called the Rademacher functions Figure : Rademacher functions r x r x r 3 x Here r k alternates + and - on the dyadic intervals of the generation k as shown in Figure Moreover, r k s are independent, identically distributed random variables with zero mean and variance one Define f n = n k= a kr k where a k } is a sequence of real numbers Then f n } is a dyadic martingale Example Let f L [0, and Q n be a dyadic interval of length on [0, Define n f n x = Q n Q n fxdx, x Q n Then f n } n= is a dyadic martingale on [0, 4 Useful results In this section, we state some useful results which will be frequently used in later chapters 4

14 Lemma If E n } is a sequence of sets on a σ algebra F with the property that E n E n+ for all n and E = n= E n, then lim n E n = E Proof: Let us define A n } as follows: A = E and A n = E n E n for n =, 3, Clearly A n F and A i A j = φ for all i j Moreover we have E n = A A A n and E = i=a i Using the disjointness of A is we have, n n E n = A i = A i and E = A i = Hence we have i= i= lim E n = lim n n A i = i= Theorem 3 For a dyadic martingale, we have i= A i = E i= A i x : f x < } as = x : Sfx < } as = x : lim f n x exists} where as = means that the sets are equal up to sets of measure zero Proof: For the proof, see [] Lemma 4 Borel-Cantelli If A n } is a sequence of events and n= P A n <, then P A n io} = 0 Proof: We first note that, A n io} = lim sup A n = n= k=n A k n is the event which occur if and only if infinite number of events A n occur i= P A n io} = P lim sup A n n = P n= k=n A k = lim P n k=n A k lim P A k n k=n = 0 5

15 Remark The Borel-Cantelli Lemma can also be stated as: Let E k } be a sequence of measurable sets in X, such that k= µe k < Then almost all x X lie in at most finitely many of the sets E k Lemma 5 Borel-Cantelli, General version If A n } is a sequence of independent events and n= P A n =, then P A n io} = Proof: We have, P A n io} = P A n io} c = P A n io} = P n= k=n A k} c = P n= k=n Ac k = lim n P k=n Ac k Clearly A c k } is a sequence of independent events as A k} is independent Then, P k=n Ac k = lim N P N = lim N k=n = lim N k=n lim N k=n = lim N exp = exp = 0 k=n Ac k N P A c k N [ P A k ] N exp P A k P A k k=n P A k k=n Hence we have P A n io} = 0 Consequently, P A n io} = 6

16 Lemma 6 Lévy s inequality If X, X, X n be independent and symmetric random variables Let S n = X + X + + X n Then for all λ > 0, P max S k λ P S n λ k n P max X k λ P S n λ k n Proof: Define A k = } x : max S jx < λ S k x j<k for k n This means k is the smallest index for which S k x λ Then using the fact that X, X,, X k independent from X k+, X k+,, X n we have, n P x : S n x λ} = P A k x : S n x λ} Thus, Similarly we get, = k= n P A k x : S n x S k x} k= n P A k P S n S k 0 k= n P A k / k= = P max S k λ k n using symmetry P S n λ P max S k λ 0 k n P S n λ P max S k λ 0 k n From 0 and 0 we have, P S n λ P max S k λ k n This gives the first result Again we define, A k = } x : max X jx < λ X j x j<k 7

17 for k n Fix k We let S 0 n = X k S n Then on A k we have, λ X k S n + S 0 n Moreover, P A k P A k S n λ} + P A k S 0 n λ} = P A k S n λ} Then summing over k we have, P A = P A S n λ} P S n λ Thus, P max X k λ P S n λ k n Lemma 7 For any λ Proof: For the proof, see [6] λ + λ e λ λ e u du λ e λ Theorem 8 Central limit theorem Let X, X, X n be a sequence of independent identically distributed random variables with finite mean µ and variance σ Define X n = /n n i= X i Then n X n µ/σ converges to a standard normal distribution Proof: For the proof, see page 36 of [3] Theorem 9 Hoeffding, 963 Let Y, Y,, Y n be independent random variables with zero mean and bounded ranges: a i Y i b i, Y i n Then for each η > 0, n η P Y i η exp n i= b i a i i= Proof: For the proof, see [9] 8

18 Theorem 0 Let X n, F n } be a submartingale and let φ be an increasing convex function defined on R If φx is integrable for every n, then φx n, F n } is also a submartingale Proof: For the proof, see [6] Lemma If X i are independent random variables with the property EX i = 0, then S n = n i= X i is a martingale and S n is a submartingale Proof: For the proof, see [6] Theorem Doob s Maximal Inequality If X n, β n is a submartingale, then for any M > 0, Proof: For the proof, see [6] P max X k M k M M EX+ n = M EmaxX n, 0 5 Origin of law of the iterated logarithm Before we discuss the origin of law of the iterated logarithm, we first give the definition of normal numbers Definition 3 Normal numbers Let us suppose that N takes values in [0, and consider its decimal and dyadic expansion as N = X n n= ; X 0 n n 0,,,, 9} N = X n n= ; X n n 0, } Now for a fixed k, 0 k 9, let ω n k N denotes the number of digits among the first N n n digits of N that are equal to k Then ωn k is the relative frequency of the digit k in the ω n k first n places and thus the limit lim N is the frequency of k in N Then the number n n N is called the normal to the base 0 if and only if this limit exists for each k and is equal to Similarly, the number N is called the normal to the base if and only it the limit 0 exists and is equal to The first law of the iterated logarithm LIL, introduced in probability theory, had its origin in attempts to perfect Borel s theorem on normal numbers Precisely, the first 9

19 LIL was introduced to obtain the exact rate of convergence in the Borel s theorem Many mathematicians obtained the different rates of convergence, but Khintchine was the one who obtained the exact rate of convergence In order to describe Khintchine s result, we state a simple form of Borel s theorem on normal numbers Theorem 4 Borel If N n t denote the number of occurrences of the digit in the first N n t n places of the binary expansion of a number t [0,, then lim = for ae t in n n Lebesgue measure So by Borel s theorem we can conclude that ae t [0, is a normal number Here n/ is the expected number of ones and the theorem gives the limit of the relative frequency of number of ones But what can be said about the deviation N n t n/? In order to answer this, we consider a special case as follows Suppose that X n takes values ± with probabilities coin tossing model We consider the unit interval with Lebesgue measure as a probability space Then we can write X n t = b n t, where b n is the n th digit in the binary expansion of t [0, Let S n = n i= X i Under this context the following results were obtained Hausdorff 93 obtained S n = On +ε ae for any ε > 0 Hardy and Littlewood 94 obtained S n = O n log n ae Khintchine 93 obtained S n = O n log log n ae In 94, Khintchine obtained the definite answer to the size of the deviation in Borel s theorem and his result is given by, Theorem 5 Khintchine If N n t denote the number of occurrences of the digit in the first n places of the binary expansion of a number t [0,, then for ae t, we have lim sup n S n n log log n = 03 0

20 This result is popularly known as Khintchine s law of the iterated logarithm LIL We note that S n t = n i= X it = n i= b it n i= = N nt n 04 Then using 04 in 03 we have, lim sup n N n t n n log log n = lim sup n N n t n = n log log n So Khintchine s LIL provides the size of the deviation in terms of expected mean and the deviation is of order n log log n Because of the factor log log n iteration of log in the deviation, Khintchine s law is popularly known as law of the iterated logarithm Borel s theorem immediately follows from the Khintchine theorem For this, we have This gives us Again Hence we have lim sup n S n t < n log log n S n n log log n = ie ae b n t n < n log log n N n t n < ie N n t n < n log log n n log log n N n t n Then taking limit as n, we have lim N n t n n log log n < n N n t = 0, so lim n n = We note that results of Hausdorff and Hardy-Littlewood also imply the conclusion of Borel s theorem For this we note that for all n, log n n so that log log n log n Consequently we have, n log log n n log n ie S n n log log n S n n log n

21 Thus, S n = O n log log n = S n = O n log n Khintchine s result on the rate of convergence is the first law of the iterated logarithm in the theory of probability A few years later, the result of Khintchine was generalized by Kolmogorov to a wide class of sequences of independent random variables We now state Kolmogorov s celebrated law of the iterated logarithm Theorem 6 Kolmogorov, 99 Let X n } n= be a sequence of independent random variables with zero mean and variance one Suppose that X n ε nn for some constants log log n ε n 0 Then for almost every ω, where S n = n X i i= lim sup n S n ω n log log n = We remark that in the above theorem, the mean of S n is zero and n is the standard deviation of S n So Kolmogorov s LIL provides the size of oscillation of partial sum of independent random variable from its expected mean and the size is approximated in terms of standard deviation Next, we apply Kolmogorov s LIL to random walks to estimate the size of the walk in the long run Consider the Rademacher functions, r k } k= Set f x = r x f x = r x + r x f n x = r x + r x + + r n x Here f n x} defines a random walk In this random walk, we move unit to the right if r i x = and to the left if r i x = Clearly, f n } satisfies all the assumptions of

22 Kolmogorov s theorem So by Kolmogorov s LIL, we have, lim sup n f n x n log log n For ε > 0, this gives us f n x + ε n log log n for n large Here, the worst bound for the function f n x is n, ie, f n x n Thus, Kolmogorov s LIL gives the sharper asymptotic estimate, f n x + ε n log log n For sufficiently large n, the factor n log log n is much smaller than n This shows that in the long run the walker will fluctuate in between n log log n and n log log n Over the years people have made many efforts to obtain an analogue of Kolmogorov s LIL in various settings in analysis Some of the existing settings are lacunary trigonometric series, martingales, harmonic functions to name just a few But the first LIL in analysis was obtained in the setting of lacunary trigonometric series Definition 7 Lacunary series A real trigonometric series with the partial sums S m = m k= a k cos n k + b k sin n k which has n k+ n k > q > is called q lacunary series In the definition, the condition n k+ n k > q > is called gap condition which states that the sequence n k } increases at least as rapidly as a geometric progression whose common ratio is bigger than Lacunary series exhibit many of the properties of partial sums of independent random variables In the modern probability theory, lacunary series are called weakly dependent random variables The law of the iterated logarithm in the setting of lacunary series was first given by Salem and Zygmund This result of Salem and Zygmund is the first law of the iterated logarithm in analysis [] Theorem 8 R Salem and A Zygmund, 950 Suppose that S m is a q lacunary series and the n k are positive integers Set Bm = m k= a k + b k and M m = max a k + k m b k Suppose also that Bm as m and S m satisfies the Kolmogorov-type B m condition: Mm K m log loge e + Bm for some sequence of numbers K m 0 Then lim sup m S m B m log log B m 3

23 for almost every T, unit circle Hence Note that π S π mxdx = 0 This means that the mean of the partial sums is zero Again, [ σ = V ars m x = π π ] S π mxdx S m xdx = π = π = π π π π π π S mxdx 0 π [ m k= a k cosn k x + b k sinn k x] dx m k= π [a k cos n k x + b k sin n k x]dx π = m a k + b k k= σ = B m = m a k + b k This shows that B m is the variance of partial sums So the theorem gives us the upper bound for the size of oscillation of partial sums from its expected mean and the order of the size depends on the size of standard deviation Salem and Zygmund assumed n k to be positive integers and they only obtained the upper bound Erdös and Gál were the first to make progress towards the other inequality They obtained the following result for a particular form of lacunary series given by the following theorem Theorem 9 Erdös and Gál, 955 Suppose S m = m k= expin k is a q lacunary series and n k are integers Then for almost every in the unit circle lim sup m k= S m m log log m = Later, M Wiess gave the complete analogue of Kolmogorov s LIL in this setting This result was the part of her PhD thesis Theorem 0 MWeiss, 959 Suppose S m = m k= a k cos n k + b k sin n k is a q lacunary series Set B m = m k= a k + b k and M m = max k m a k + b k Suppose 4

24 also that B m as m and S m satisfies the Kolmogorov-type condition: M m K m B m log loge e + B m for some sequence of numbers K m 0 Then lim sup m for almost every in the unit circle S m B m log log B m = There is another type of LIL in the case of independent random variables introduced by Kai Lal Chung Theorem Chung, 948 Let X n ; n } be a sequence of independent identically distributed random variables with common distribution F with zero mean and variance σ, log log n and with finite third moment E X 3 < Then lim inf max n n S j = σπ with j n 8 probability Next, we discuss another law of the iterated logarithm introduced by Salem and Zygmund In this LIL, they considered tail sums of the lacunary series instead of n th partial sums Theorem R Salem and A Zygmund, 950 Suppose a lacunary series S N = k=n a k cos n k + b k sin n k where c k = a k + b k satisfies k= c k < Define B N = k=n k c and M N = max c k Suppose that B B N < and that M N K N k N log log B N for some sequence of numbers K N 0 as N Then lim sup N for almost every in the unit circle S N B N log log B N This result is popularly known as tail law of the iterated logarithm We remark that the condition k= c k < says that the given lacunary series converges ae and S N = N a k cos n k + b k sin n k a k cos n k + b k sin n k This shows that the tail LIL gives k= k= the rate of convergence of partial sums of lacunary series to its limit function Furthermore, the rate of convergence depends upon the standard deviation of the tail sums 5

25 6 Organization of the thesis The purpose of this thesis is to obtain an analogue of Salem-Zygmund s tail law of the iterated logarithm in various contexts in analysis The various contexts are Rademacher functions, dyadic martingales, independent random variables, and lacunary trigonometric series We first establish the tail law of the iterated logarithm for sums of Rademacher functions which are nicely behaved dyadic martingales Employing the idea from the Rademacher functions, we then derive the tail law of the iterated logarithm for dyadic martingales and then obtain the tail law of the iterated logarithm for independent random variables and lacunary series The thesis is organized as below In chapter, we derive some standard inequalities which will be used in later chapters and derive the martingale analogue of Kolmogorov s law of the iterated logarithm Chapter 3 begins with the derivation of the tail law of the iterated logarithm for sums of Rademacher functions We then derive the tail LIL for dyadic martingales and construct an example of a dyadic martingale which does not follow the tail LIL In this chapter, we only focus on the upper bound in the LIL for these functions In chapter 4, we obtain a lower bound in the tail LIL for sums of Rademacher functions We also introduce the tail law of the iterated logarithm for sums of independent random variables and obtain a lower bound for it In chapter 5, we obtain a lower bound in the tail law of the iterated logarithm for dyadic martingales and finally in chapter 6, we obtain the lower bound in the tail law of the iterated logarithm for lacunary series introduced by Salem and Zygmund 6

26 Chapter Law of the iterated logarithm In this chapter, we first derive two useful martingale inequalities and then obtain an analogue of Kolmogorov s law of the iterated logarithm in the case of dyadic martingales The martingale analogue of Kolmogorov s law of the iterated logarithm was first derived by W Stout Stout obtained the martingale analogue using a probabilistic approach We will derive it using the harmonic analysis approach Martingales inequalities We first prove a Lemma, called Rubin s Lemma which will be used in our martingale inequalities The proof of this lemma can also be found in [0], [], and [4] Lemma 3 Rubin For a dyadic martingale f n }, f 0 = 0 0 exp f n x S nfx dx Proof: We claim that gn = n exp d k x 0 k=0 n d kx dx k=0 is a decreasing function of n Let Q nj be an arbitrary n th generation dyadic interval We 7

27 have n k=0 d kx = f n and f n is constant on Q nj Using this we have, n n+ gn + = exp d k x n+ d j=0 Q nj kx dx k=0 k=0 n n = exp d k x n d j=0 Q nj kx exp d n+ x d n+x dx k=0 k=0 [ n n ] = exp d k x n d kx exp d n+ x Q nj d n+x dx Let Q n+j j=0 k=0 k=0 Q nj and Q n+j be the dyadic subintervals of Q nj Suppose d n+ takes the value α on Q n+j Then by the expectation condition, d n+ takes the value α on Q n+j This gives, Q nj exp d n+ x d n+x dx = = Q n+j exp α α dx + [ exp α α = exp = exp α α + exp e α + e α cosh α Now using the elementary fact that cosh x e x, we have [ n n ] gn + exp d k x n d kx exp j=0 k=0 k=0 Q nj [ n n ] = exp d k x n d kx Q nj j=0 k=0 k=0 Q nj n n = exp d k x n d Q nj kx dx j=0 = gn k=0 k=0 Q n+j α α ] n+ n+ α exp exp α α dx n+ α n+ Let Q and Q be the dyadic subintervals of Q 0 Assume that d takes value on Q so that it takes value on Q 8

28 g = = 0 0 exp d x d x dx exp dx + exp dx = exp + exp = exp e + e = exp cosh exp exp = Since gn is decreasing and g we conclude, n exp d k x n d kx dx 0 k=0 k=0 Hence, This completes the proof of Rubin s lemma 0 exp f n x S nfx dx Note that if we rescale the sequence f n } by λ, then the Lemma gives, 0 exp λf n x λ Snfx dx This shows that the Rubin s lemma is an inhomogeneous type inequality Now we prove our first martingale inequality Lemma 4 For a dyadic martingale f n } and λ > 0 we have λ x [0, : sup f m x > λ} 6 exp m Sf 9

29 Proof: Fix n Let λ > 0, γ > 0 Then and so on Hence for every m n, f n x = f n ydy, x Q n, Q n = Q n Q n n f n x = Q n Q n f n ydy, x Q n, Q n = n f m x = f n ydy, x Q m, Q m = Q m Q m m Fix x Then sup f m x M f n x where Mf n is the Hardy-Littlewood maximal m n function of f n Then using Jensen s inequality we have, expγ f m x = exp γ Q m f n yd Q m Me γ fmx x y Q m Q m expγ f n y dy Then the Hardy-Littlewood maximal estimate gives, } x [0, : sup f m x > λ} = x [0, : sup e γ fmx > e γλ m n m n x [0, : Me γ fm x > e γλ} 3 e γλ = 3 e expγ f n y dy 0 γ exp γλ S nf 0 γ 3 exp eγλ S nf 0 exp γ f n y γ S nf dy dy exp γ f n y γ S nfy 0

30 Applying the Rubin s Lemma we have, 0 = = exp γ f n y γ exp y:f ny 0} y:f ny 0} 0 + = exp S nf dy γ f n y γ S nfy dy + S nfy dy + dy + exp exp γf n y γ γf n y γ S nfy 0 y:f ny<0} y:f ny<0} exp γ f n y γ S nfy dy exp γf n y γ S nfy dy dy γf n y γ S nfy Thus, Choose γ = } x [0, : sup f m x > λ 6 γ exp m n eγλ S nf λ Then we have, S n f 6 λ x [0, : sup f m x > λ} S n f exp m n λ exp S n f 4 S nf λ 6 exp S n f For the dyadic martingale f n }, S nfx = n d kx S fx = k= d kx k= This gives, S n f Sf Consequently, S n f Sf So we have, λ x [0, : sup f m x > λ} 6 exp m n Sf

31 Define E n := x [0, : } sup f m x > λ m n and E := x [0, : sup f m x > λ m Clearly E n E n+ and E = k=e k Then we have, lim n E n = E See Lemma for the proof Thus, So, x [0, : sup f m x > λ} lim m n x [0, : sup f m x > λ} m n λ lim 6 exp n S n f λ = 6 exp S n f λ x [0, : sup f m x > λ} 6 exp m Sf This completes the proof of the first martingale inequality sums Now using the above martingale inequality, we prove a martingale inequality for tail Lemma 5 For a dyadic martingale f n }, with λ > 0 and, n fixed positive integer we have, λ x [0, : sup fx f m x > λ} exp m n 8 S nf } Proof: Fix n Define a sequence g m } as follows, g m x = 0, if m n; f m x f n x, if m > n We first show that g m } is a dyadic martingale Clearly for every m, g m is measurable with respect to the sigma algebra F m Let m > n Then using the fact that f m is constant on the cube Q m we have,

32 Eg m+ F m x = [f m+ x f n x]dx Q m Q m = f m+ xdx f n xdx Q m Q m Q m Q m = f m+ xdx f n x Q m Q m = f m x f n x = g m x Thus we have Eg m+ F m = g m This shows that g m } is a martingale Then applying Lemma 4 for this martingale, we get λ x [0, : sup g m x > λ} 6 exp m Sg But, g m x = 0 for m n Hence, λ x [0, : sup g m x > λ} 6 exp m n Sg Again, S gx = d kx = k=0 = = = = [g k+ x g k x] k=0 [g k+ x g k x] k=n [f k+ x f n x f k x + f n x] k=n k=n+ k=n+ [f k+ x f k x] d kx = S n fx This gives, λ x [0, : sup g m x > λ} 6 exp m n S nf 3

33 ie Clearly, So we have, λ x [0, : sup f m x f n x > λ} 6 exp m n S nf x : fx f n x > λ} x : sup f m x f n x > λ} m n 0 x : fx f n x > λ} x : sup f m x f n x > λ} m n Consequently, By the triangle inequality we have, λ x : fx f n x > λ} 6 exp 0 S nf sup m n This gives, } x : sup fx f m x > λ m n fx f m x sup fx f n x + f n x f m x m n = fx f n x + sup f n x f m x m n x : sup fx f n x > λ } x : sup f n x f m x > λ } m n m n Therefore, x : sup fx f m x > λ} x : fx f m x > λ } + m n x : sup f n x f m x > λ } m n Then using 0 and 0 in the above inequality we get, λ x : sup fx f m x > λ} 6 exp m n S nf λ = exp 8 S nf + 6 exp λ S nf Thus, λ x : sup fx f m x > λ} exp m n 8 S nf This completes the proof of our second martingale inequality 4

34 Law of the iterated logarithm for dyadic martingales Burkholder and Gundy proved See Theorem 3 x : Sfx < } as = x : lim f n exists} where as = means the sets are equal upto a set of measure zero From this result, we observe that dyadic martingales f n } behave asymptotically well on the set x : Sfx < } But what can be said about the asymptotic behavior of dyadic martingales on the complement of the given set? Its behavior is quit pathological on the set x : Sfx = }; in particular it is unbounded ae on this set But it is possible to obtain the size of growth of f n on the set x : Sfx = }? The rate of growth of f n on x : Sfx = } is precisely given by the martingale analogue of Kolmogorov s law of the iterated logarithm W Stout proved the law of the iterated logarithm for martingales using a probabilistic approach Here we derive the law of the iterated logarithm for dyadic martingales using a harmonic analysis approach Theorem 6 If f n } n=0 is a dyadic martingale on [0, then, lim sup n f n x S n fx log log S n fx almost everywhere on the set where f n } is unbounded Proof: Let > and δ > 0 We note that for every x [0,, we have either S n fx > k for some n or S n fx k, for every n, and thus, Sfx k We define stopping time as; So by stopping time, γ k S γk fx k Define, γ k x = minn : Sn+ fx > k, if Sfx k is the smallest index such that S γk +fx > k This means f n x = f n γk x = f x, f x,, f γk x, f γk x,, for γ k f x, f x, f 3 x,, if γ k = 5

35 We first show that S f k So for n < γ k x, we have S f n x = Sf n x Sf γk x k Again if n γ k x, then S f n x = Sf γk x k Thus, n S f n x k Then, lim S f n x k So we have S f k n Choose λ = +δ k log log k Then using Lemma 4 for the dyadic martingale f n } with the chosen λ, we get } x [0, : sup f n x > + δ k + δ k log log k log log k 6 exp n Sf + δ k log log k 6 exp k = 6 exp + δ k log log k = 6 exp logk log +δ = 6k log +δ = 6 k log +δ Summing over all k, we have } x [0, : sup f n x > + δ k 6 log log k n log +δ k +δ k= k= 6 = log +δ k +δ < Then by Borel-Cantelli Lemma Lemma 4 we have for ae x, sup f n x + δ k log log k n for sufficiently large k, say, k M, M depends on x Thus for ae x, we have, sup f n γk xx + δ k log log k n for sufficiently large k M We choose x such that f n x is unbounded Then from the Theorem 3 we have, x : Sfx < } ae = x : f n x converges} 6 k=

36 So we have Sfx = Then γ x γ x γ 3 x ie for every i, γ i x < Let n γ M Then choose k such that γ k x < n γ k+ x Here, γ k x < n gives γ k x n Thus, S n fx = S n + fx > k Using this, we have So, lim sup n We show, f n x sup m γ k+ f m γk+ x f n x sup f m γk+ x m S n fx loglog S n fx Let X = logs n fx Then lim sup n Therefore for ae x, + δ k+ log log k+ = + δ k loglog k + log < + δs n fx loglog S n fx + log lim sup n < + δ lim sup n loglog S n fx + log loglog S n fx loglog S n fx + log loglog S n fx = = lim sup X loglog S n fx + log loglog S n fx = logx + log log X Letting we get, lim sup n lim sup n f n x S n fx log log S n fx < + δ f n x S n fx log log S n fx + δ This can be done for every δ > 0 Hence we have for ae x, lim sup n f n x S n fx log log S n fx This completes the proof of the law of the iterated logarithm for dyadic martingales 7

37 Chapter 3 The tail law of the iterated logarithm In this chapter, we first establish the tail law of the iterated logarithm for sums of Rademacher functions Sums of Rademacher functions are nicely behaved dyadic martingales We then derive the tail law of the iterated logarithm for dyadic martingales Moreover, with the help of an example we will show that the tail law of the iterated logarithm is not true in general 3 The tail law of the iterated logarithm for sums of Rademacher functions We first prove a lemma which will be used in the proof of the tail LIL for sums of Rademacher functions Lemma 7 Let f n = n k= a kr k, f = k= a kr k where a k } is a sequence of numbers and r k } is a sequence of Rademacher functions Then for a fixed n and λ > 0 we have, λ x : sup fx f m x > λ} 4 exp m n S nf Proof: Let d i = f i f i Then d i = i k= a kr k i k= a kr k = a i r i Here, each d i has mean 0 and variance Moreover, they are independent and symmetric random variables So using Lévy s inequality Lemma 6, Chapter, we have P max j j n i= d j > λ P n k= d k > λ 8

38 Let N >> n Then we have, P max j 0 j N n i=0 d N i > λ Thus, P N n i=0 d N i > λ P max d N, d N +d N,, d N +d N + +d n+ > λ P d N +d N + +d n+ > λ This gives, P max f N f N, f N f N,, f N f n > λ P f N f n > λ ie } x [0, : max f Nx f m x > λ x : f N x f n x > λ} N m n Using the fact that sup k a k > λ sup k a k > λ or sup k a k > λ we have, } x [0, : max f Nx f m x > λ N m n } = x [0, : max f } Nx f m x N m n x > λ [0, : max f Nx + f m x N m n > λ Then, x [0, : max f Nx f m x > λ} N m n } } x [0, : max f Nx f m x N m n > λ + x [0, : max f Nx f m x N m n > λ < x : f N x f n x > λ} + x : f N x f n x > λ} = x : f N x f n x > λ} Thus, } x [0, : max f Nx f m x > λ x : f N x f n x > λ} N m n Now using equation 0 of Chapter we have, λ x [0, : sup f m x f n x > λ} 6 exp m n S nf 9

39 Thus, Clearly, Therefore, Let E N = x : f N x f n x > λ} x [0, : sup f m x f n x > λ} m n λ x [0, : f N x f n x > λ} 6 exp S nf λ x [0, : sup f N x f m x > λ} exp N m n S nf x [0, : } sup f N x f m x > λ N m n and E = k= E k Clearly E N E N+ Then E = lim N E N See Lemma, Chapter for the proof Next we show, } x [0, : sup fx f m x > λ m n E Let x be such that sup fx f m x > λ Then for sufficiently large N we have, m n sup N m n f N x f m x > λ This means x E N for sufficiently large N so that x E Then, x [0, : sup fx f m x > λ} E m n = lim E N N = lim N x [0, : sup f N x f m x > λ} N m n λ lim exp N S nf λ = exp S nf Thus, λ x [0, : sup fx f m x > λ} exp m n S nf This completes the proof of the Lemma 30

40 Theorem 8 Tail LIL for Rademacher functions Let r k } k= be the sequence of Rademacher functions and a n } n= be a sequence with n= a n < Set ft = k= a kr k t, f n t = n k= a kr k t, S n ft = j=n+ a k Then, for ae t lim sup n ft f n t S n ft log log S n ft Proof: We first show that f n } n= is a dyadic martingale For this we note that for i n, a i r i is measurable with respect to F n and so is the sum n i= a ir i = f n Again for Q n = n, we have Ef n+ F n = Q n = Q n = Q n = Q n f n+ xdx Q n n+ a k r k xdx = Q n n = = k= Q n k= Q n [ n ] a k r k x + a n+ r n+ x dx k= Q n k= Q n k= a k r k x Q n n a k r k x k= = f n Let > Define n n, n k by n k = min n a k r k xdx + a n+ r n+ xdx Q n n a k r k xdx + 0 n : j=n+ Q n dx a j < k Using the previous Lemma Lemma 7 for a fixed m we have, λ t : sup ft f n t > λ} exp n m S mf 3

41 Then using the above estimate for n k we have, t : sup ft f n t > λ} exp n n k But S ft = j=n k + f j t f j t = j=n k + λ S n k f a j r j t = j=n k + So, λ t : sup ft f n t > λ} exp n n k j=n k + a j We choose λ = + ε log log k k where ε > 0 Then, from the above estimate, we have } + ε log log k t : sup ft f n t > + ε log log k exp n n k k k j=n k + a j Using j=n k + a j < we get, k } + ε log log k t : sup ft f n t > + ε log log k exp n n k k k a j k = exp + ε log log k = exp logk log +ε = k log +ε = log +ε k +ε Consequently, } t : sup ft f n t > + ε log log k n n k= k k < k= log +ε = log +ε < k +ε k +ε k= 3

42 So by Borel-Cantelli Lemma Lemma 4, Chapter for ae t, sup ft f n t + ε n n k log log k k 0 for sufficiently large k, say k M such that M depends on t Fix t Choose n n M Then k M such that n k n < n k+ Now by the definition of n k+, we have S + ft < k+ But n < n k+ so that S n ft Thus, k+ Again n k n implies Thus, Then using 0 in 0, we have Thus for ae t we have, j=n+ j=n+ k+ a j k+ a j < k j=n+ a j < k 0 ft f n t sup ft f m t m n k + ε log log k k = + ε log log k k+ < + ε a j log log j=n+ a j lim sup n Letting, we get j=n+ a j j=n+ ft f n t < + ε log log j=n+ a j 33

43 lim sup n j=n+ a j This is true ε > 0 Hence for ae t, lim sup n j=n+ ft f n t < + ε log log j=n+ a j a j ft f n t log log j=n+ a j This completes the proof of the tail law of the iterated logarithm for sums of Rademacher functions Remark In the above theorem, we have S fx = n= a n < Then by Theorem 3, Chapter, lim f n x exists So the tail law of the iterated logarithm gives the rate of convergence of the sequence f n } to its limit function f and rate of convergence depends on the tail sums of the square function 3 The tail law of the iterated logarithm for dyadic martingales In this section, we employ the idea from the Rademacher case to obtain the tail law of the iterated logarithm for dyadic martingales Moreover, we will later note that the tail law of the iterated logarithm in not true in general which will be justified by an example Theorem 9 Tail LIL for dyadic martingales Let f n } n=0 be a dyadic martingale Assume that there exists a constant C < such that S nfx S nfy C x, y I nj for n = [ j,, 3,, j 0,,, 3,, n } where I nj =, j + Then n n for ae x lim sup n fx f n x S n fx log log S n fx C 34

44 Proof: Let > Define functions γ γ by γ k x = min n : x I nj for some j,, 3, n } and y I nj, S nfy < } k Now by Lemma 5, Chapter for each I nj we have, λ I nj y I nj : sup fy f n y > λ} exp n m 8 S m f Inj Now using the above estimate for γ k y, we have, Here, I nj y I nj : sup fy f n y > λ} exp n γ k y λ 8 S γk y f I nj 03 So S γ k yfy Inj < k S γ k yf Inj k λ S γ k y f λ k I nj λ λ k exp 8 S γ k y f exp 04 I nj 8 Then using 04 in 03, we get y I nj : sup f n y fy > λ} I nj exp n γ k y λ k Now summing over all such I nj we get, y [0, : sup λ k f n y fy > λ} exp 8 k= n γ k y k= 8 05 Summing over all over all generations we have, y [0, : sup f n y fy > λ} n γ k y λ k exp 06 8 Choose λ = + ε log log k k where ε > 0 Then using 06 for the chosen λ, we have 35

45 Thus, k= y [0, : k= + ε exp k= sup f n y fy > + ε n γ k y k log log k k 8 + ε log log k = exp 4 k= = exp logk log +ε 4 = k= k= = log +ε 4 < k log +ε 4 k= k +ε 4 y [0, : sup f n y fy > + ε n γ k y log log k k log log k k } } < Hence, by Borel-Cantelli Lemma Lemma 4, Chapter for ae y we have, sup f n y fy + ε n γ k y log log k k 07 for sufficiently large k, say k M, M depends on y Fix y Choose n n M, then j such that y I nj and k M such that γ k y n < γ k+ y By the definition of γ k we have, S γ k y fy < k and γ ky n This gives, S γ nyfy < k 08 Again by the definition of γ k+, S γ k+ y fy < and n < γ k+y So for some y k+ 0 I nj, S nfy 0 But y, y k+ 0 I nj, so by assumption S n fy 0 S C Thus, n fy CS nfy S nfy 0 k+ 36

46 ie Combining 08 and 09 we have, CS nfy 09 k+ C k+ S nfy < k 00 Using 00 in 07, we have f n y fy sup m γ k y f m y fy + ε log log k k log log = + ε k log log k k+ log log k + ε C log k + log log S nfy log log S n fy log k + log log f n y fy S n fy log log S n fy Also we know that as n so does k Here, lim sup n So for ae y we have, f n y fy S n fy log log S n fy lim sup n lim sup k log k + log log + εc log k + log log + εc log k + log log log k + log log = f n y fy S n fy log log S n fy lim sup k < + εc log k + log log log k + log log 37

47 Letting we get, lim sup n f n y fy S n fy log log S n fy This is true for ε > 0 Thus for ae y we have, C + ε lim sup n f n y fy S n fy log log S n fy C This proves the tail law of the iterated logarithm for dyadic martingales Remark 3 From the assumption, we get Sfx < for ae x This shows that the sequence f n x} converges by Theorem 3 Thus the tail law of the iterated logarithm gives the rate of convergence of dyadic martingales f n } to its limit function f Moreover, the rate of convergence depends on the tail sums of martingale square function Corollary 30 Let f n } n=0 be a dyadic martingale Fix > Define stopping times, n k x = min n : x I nj for some j,, 3, n } and y I nj, S nfy < } k Then for the sequence of stopping times n k x, for ae x lim sup k S x f xx fx [ fx log log S x fx ] < 3 Proof: We first prove the following estimate for λ > 0, η > 0, x [0, : fx f n x > λ, From equation 0 of Chapter we have, } S nfx < ηλ 6 exp 0 η λ x : fx f n x > λ} 6 exp S nf 38

48 Here S nfx < ηλ gives S nf η λ So, S nf x [0, : fx f n x > λ, η λ Then } S nfx λ < ηλ 6 exp S nf λ 6 exp η λ = 6 exp η Choose λ = + ε log log l l and η = + ε log log l where > and ε > 0 Then using 0 we have, } x [0, : fx f n x > + ε log log l, S nfx < l l + ε log log l 6 exp + ε = 6 exp logl log = 6l log = = 6 l log +ε 6 log +ε + ε +ε l Choose ε = + ε 3 Then we have = 3 Thus, x [0, : fx f n x > + ε log log l, S nfx < l 3 6 log l 3 Let gx = = C l 3 say l x log log x Clearly gx is an increasing function So for l S n fx, 39 }

49 we have S n fx log log S n fx Now using 0, we have x [0, : fx f n x > + ε = x [0, : fx f n x > + ε l=k+ x [0, : fx f n x > + ε l=k+ = x [0, : fx f n x > l=k+ x [0, : fx f n x > l=k+ l=k+ Clearly, C l 3 l=k+ S n fx log log l log log l 0 S n fx S n fx log log } n fx, S nfx < l S l log log l, S nfx < l + ε log log l, S nfx < } l l + ε l log log l, S nfx < l } [ l 3 k x dx = 3 x Thus, x [0, : fx f n x > + ε ] k = k S n fx log log This can be done for every n k x So summing over all k we have, S n fx } } C k l } } x [0, : fx f xx > + ε S x fx log log S k= x fx C k k= = C k < k= 40

50 So, by Borel-Cantelli Lemma 4, Chapter for ae x, there exists M which depends on x such that for every k M, But ε = 3 So, fx f xx + ε S x fx f xx [ fx log log S x fx log log S x fx S n k x fx ] 3 We note that as n so does k Letting we get for ae x, lim sup k S x fx f xx [ fx log log S n k x fx ] 3 Remark 4 This is true for every stopping time But we can not estimate the behavior of the limsup in between any two stopping times as two consecutive stopping times might be significantly different Next, let f be an integrable function such that f x is continuous and x, 0 < m f x M Let us define f n x = Ef F n x where F n is the σ algebra generated by the dyadic intervals of length n on [0, Clearly 4

51 the sequence f n } defines a dyadic martingale Then, d n x = f n x f n x = fydy fydy Q n Q n Q n Q n Q n Q n = fydy fydy fydy Q n Q n Q n Q n Q n Q n Q n = Q n Q n = fydy fydy fu + Q n du using y = u + Q n Q n Q n Q n Q n Q n Q n = fydy fydy fu + Q n du Q n Q n Q n Q n Q n Q n = fydy fu + Q n du Q n Q n Q n Q n = [fy fy + Q n ] dy Q n Q n Now by mean value theorem there exists z such that fy fy + Q n = f z Q n But f z M Thus fy fy + Q n M Q n Then we have, Now using f x m, we have d n x M Q n dy = Q n Q n Q n M Q n Q n = M 03 n+ d n x = [fy fy + Q n ] dy Q n Q n = f z Q n dy Q n Q n = m n+ Q n mdy Hence we have, Now d n x m 04 n+ fx f n x = k=n+ d kx k=n+ d kx 4

52 From 03 we have Then we get Again using d k x Let us take gx = g fx f n x m, we have k+ S n fx = k=n+ d k x M k+ d kx k=n+ k=n+ M = M k+ n+ m k+ = m 05 3n+ x log log Clearly gx is an increasing function So we have, x n fx S S n fx log log m g 3 n+ S n fx m 3n+ log log 3n+ m So we have S n fx log log S n fx Hence using 06 we have, m 3n+ log log 3n+ m 06 lim sup n f n y fy S n fy log log S n fy lim sup n = lim sup n m M n+ 3n+ log log 3n+ m 3M m log log 3n+ m 0 as n This shows that there is no need to use our theorem to find the limit of the quotient for functions with continuous and bounded derivatives as it is trivial in this case Clearly we do have functions fx = n= a nr n x for which limsup in law of the iterated logarithm is nontrivial where r n } is sequence of Rademacher functions We want to look for functions for Lipα type which is slightly more general than differentiable function 43

53 Next, we show that limsup is nontrivial for functions Lipα for < α < For this we consider fx = n= a n sin n x Clearly, the series satisfies the gap condition So it is a lacunary series If a n <, then it converges ae We choose a n such that fx Lipα In order to choose a n we recall a Theorem from [] Theorem 3 For the function fx for which the Fourier series is lacunary to belong to the class Lipα 0 < α < it is necessary and sufficient that its Fourier coefficients are of order On α We choose a n =, so that fx = k α k= sin k πx Let us construct martingales k α from the given function as follows f n x = Ef F n = Q n Q n k= k α sink πxdx So we have, f n x = Q n Q n k α sink πxdx k= = [ n Q n k α sink πx + = Q n = Q n Q n k= n Q n k= n k= k=n+ k α sink πxdx + 0 Q n k α sink πxdx ] k α sink πx dx By mean value theorem there exits c such that, sin k πx sin k πy = k πx y cos k πc 07 44

54 Hence using 07 we have, fx f n x Here k=n+ x ydy = Q n Hence, k α sink πx = fx f n x y:y>x} fx f n x fx n k α sink πx n Q n k α sink πxdx k= n = k α sink πx Q n k= k= = n Q n k α sink πxdy = Q n = Q n Q n x y dy k=n+ k= Q n n Q n k= n Q n k= n Q n k= x+ Qn x Q n n k= k α [sink πx sin k πy]dy k α k πx y cos k πcdy k α k πx ydy x ydy = k α sink πx Q n n = k= = π n n n k= k α k π n k= = π n l k= k k α Q n k α sink πydy [ ] x+ Qn x y = x Q n k α k π Q n k k α + n k=l+ k k α We choose l such that l+ = n n α so that log l+ = log n n α This shows that l = n + α log n 45

55 Then using α log n n for large n, we have n l k n k = k α n k + α k= k= = n l+ + n n k k α k=l+ k=l+ n n α log n + n α + n n+ l + α n α + l + α k k α n + α n + α n α log n α n + α n α = + α+ n α = C α n α say n k=l+ k We have, Next we show for some constant C lim sup N k=n k=n sin k πx k α k α log log k=n d m x C α m α k α = 46

56 d m x = f m x f m x = [ sin k πx sin k πx + Q Q m Q m k α m ] dx k= [ m = Q m Q m k α sink πx sin k πx + Q m + k= k=m+ [ m ] = Q m Q m k α sink πx sin k πx + Q m dx k= m Q m k x + Q m x k π cos k πc dx Using MVT α Q m = m k k α k= Q m k= m Q m k= π m + α+ m α = C α m α So Define gx = This gives, lim sup N k α Q m k πdx d Cα kx k α k=n+ k=n+ x log log x Clearly gx is an increasing function So k=n+ g d kx log log k=n+ d kx g k=n+ lim sup N k=n d k x C α k α k=n+ ] k α sink πx sin k πx + Q m Cα k log log α k=n+ C α k α dx 47

57 Then, lim sup N = lim sup N lim sup N lim sup N lim sup N k=n+ fx f N x d kx log log fx f N x lim sup N k=n d k x k=n+ k=n+ fx f N x sin k πx + k α k=n+ sin k πx k α Cα k log log α C α k α k=n+ k=n+ k=n+ sin k πx fx f k α N x k=n+ k=n+ k=n+ k=n+ k=n+ k=n+ k α sin k πx Cα N α Cα k log log α Cα k log log α k=n+ sin k πx k α Cα k log log α k=n+ C α k α C α k α k=n+ lim sup N C α k α Cα k log log α sin k πx k α k=n+ C α N α k=n+ Cα k log log α k=n+ C α k α C α k α But This gives, k=n+ C α k α = α N α N N α lim sup N k=n+ C α N α Cα k log log α k=n+ C α k α = lim sup N C α N α N α N log log N α N = lim sup C α N N log log N α N = 0 48

58 Thus we have, lim sup N k=n+ fx f N x d kx log log lim sup N k=n d k x = C α lim sup N k=n+ k=n+ sin k πx k α Cα k log log α k=n+ k=n+ k=n+ sin k πx k α k log log α C α k α k=n+ C α k α But lim sup N k=n+ k=n+ k log log α k=n+ k log log α k=n+ Using 08 together with the assumption, we get lim sup N k=n+ k=n+ sin k πx k α k log log α k=n+ C α k α k α C α k α = 08 = Hence we have, lim sup N k=n+ fx f N x d kx log log k=n d k x C α This shows that there are functions in Lipα with < α < for which law of the iterated logarithm is nontrivial We note that LIL is trivial for functions with continuous and bounded derivatives We showed that there are functions in Lipα with < α < doe which it is nontrivial Note that fx is differentiable ae for f Lipα α = This means the gap is very narrow Next, we show that the tail law of the iterated logarithm is not true in general with help of an example 49