Inequalities Associated to a Sequence of Dyadic Martingales

We first 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. Now we give the definition of dyadic martingales. For this let


Introduction
We first 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. Now we give the definition of dyadic martingales. For this let n denote the family of dyadic subintervals of the unit interval [0, 1) of the form 1 , 22 n n jj        where n = 0, 1, 2  and j = 0, 1,  2 n -1.
Definition 1.1 (Dyadic Martingale) (Bañuelos and Moore, 1999) A dyadic martingale is a sequence of integrable functions,   0 n n f   from [0, 1) such that: (i) For every n, fn is n F -measurable where n F is the algebra generated by dyadic intervals of the form   means the sets are equal upto a set of measure zero. From this result, we observe that dyadic martingales {fn} behave asymptotically well on the set {x: Sf(x) < }. But what can be said about the asymptotic behavior of dyadic martingales on the complement of this set? Its behavior is quite pathological on the set {x: Sf(x) = }. In particular it is unbounded a.e. on this set. In order to study the asymptotic behavior of the sequence of dyadic martingales, the martingales inequalities are helpful. These inequalities provide sub-Gaussian type estimates for the growth of the dyadic martingales. We derive these estimates for a regular sequence and a tail sequence of dyadic martingales. Asymptotic behavior of the martingales is studied through the law of the iterated logarithm of martingales (Stout, 1970). There is law of the iterated logarithm for various other contexts such as for harmonic functions, independent random variables, lacunary trigonometric series (Ghimire and Moore, 2014;Bañuelos et al., 1988). We now state our main results:

Preliminaries
We first fix some notations, give some definitions which will be used in the course of the proof.

Definition 2.1
For a dyadic martingale, The quadratic characteristics or square function: 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. From the definition, we note that for any x, yQn, we have . For more about martingales (Neveu and Speed, 1975).

Definition 2.2 (Hardy-Littlewood Maximal Function)
Let   pn fL  , 1  p  . Then Hardy-Littlewood Maximal function associated to f, denoted by Mf, is defined as: B(x, r) is the ball with center at x and radius r.

Proof of the Main Results
We first prove a Lemma. This Lemma is also known as Rubin's Lemma (Pipher, 1993). The proof of this Lemma can also be found in (Chang et al., 1985). Here we give a proof of the Lemma using a different approach. Our proof is more analytic than the original probabilistic approach. We will use this Lemma in the proof of our inequalities.

Proof of Lemma 3
We claim that: is a decreasing function of n, Let Qnj be an arbitrary nth generation dyadic interval. We have and fn is constant on Qnj. Using this we have: Since g(n) is decreasing and g(1)  1 we conclude: