One-sided law of the iterated logarithm for dyadic martingale using sub-gaussian estimates

: The martingale analogue of Kolmogorov’s law of the iterated logarithm was obtained by W. Stout using probabilistic approach. In this paper, we give a new proof of one side of the same law of the iterated logarithm for dyadic martingale using subgaussian type estimates and Borel-Cantelli Lemma.


Introduction
K olmogorov's law of the iterated logarithm (LIL) for the sequence of independent random variables is in the words of K. L. Chung, "a crowning achievement in classical probability theory". We first begin with Kolmogorov's celebrated law of the iterated logarithm. This beautiful law of the iterated logarithm result of Kolmogorov was first proved by Khintchine [2] for Bernoulli random variables. Khintchine obtained this result while improvising the efforts of Hausdorff (1913), Hardy and Littlewood (1914) and Steinhaus (1922) to obtain the exact rate of convergence in Borel's Theorem on normal numbers. Over the years, people have obtained the analog of the Kolmogorov's result in various settings in analysis. Some of the existing settings are lacunary trigonometric series, martingales, harmonic functions, Bloch functions etc. Readers are referred to a survey article by Bingham [3] which has more than 400 references on the law of the iterated logarithm. Salem and Zygmund [4] obtained the analogue of Kolmogorov's LIL in the context of lacunary trigonometric series and their result is the first LIL in analysis. Moreover, Salem and Zygmund [4] also introduced a law of the iterated logarithm for the tail sums of lacunary trigonometric series, known as tail LIL. The tail LIL of lacunary series was then completed by Ghimire and Moore [5].
In 1970, Stout [6] obtained a martingale version of Kolmogorov's LIL where he used the probabilistic approach. In this paper, we prove one side of the same law of the iterated logarithm for dyadic martingales using a different approach. Precisely, we use the harmonic analysis approach and easily obtain the upper bound. In the proof, we make the use of a subgaussian type estimate and Borel-Cantelli lemma. Our main result is:

Preliminaries
We first fix some notations, give some definitions and state some lemmas which will be used in the course of the proof. Let D n denote the family of dyadic subintervals of the unit interval [0, 1) of the form j 2 n , j+1 2 n , where n = 0, 1, 2 · · · and j = 0, 1, · · · 2 n − 1.
(ii) and the following conditional expectation condition holds Next, we define Hardy-Littlewood maximal function: Then M f is called the Hardy-Littlewood maximal function of f . Here |B(x, r)| denotes the measure of the ball centered at x and of radius r.
Let m denote the Lebesgue measure on R.
Lemma 1 (Borel-Cantelli [7]). Let {E k } ∞ k=1 be a countable collections of measurable sets for which ∑ ∞ k=1 m(E k ) < ∞. Then almost all x ∈ R belong to at most finitely many of the sets E ′ k s.
Next, we obtain an estimate for the sequence of dyadic martingales. This estimate will be used in the proof of a lemma. The estimate is stated as a lemma below: This estimate was originally obtained by Chang et al., [8] using the probabilistic approach. Recently, S. Ghimire also obtained the same estimate using the analytic approach. Please refer [9] for the detail.

Remark 1.
Note that if we rescale the sequence { f n } by λ, then Lemma 2 gives, This shows that the above inequality is inhomogeneous type. We will make the use of this form in the proof the lemma that follows.
With the help of Lemma 2, we now obtain a subgaussian type estimate related to dyadic martingales. This estimate plays the central role in the proof of our main result. The proof of the estimate can be found in [9]. We also revisit the same proof here. The estimate is given as a lemma below; Lemma 3. For a dyadic martingale { f n } and λ > 0, we have Proof. Fix n. Let λ > 0, γ > 0. Then for every m ≤ n, , where M f n is the Hardy-Littlewood maximal function of f n . Then using Jensen's inequality, we have Using the Hardy-Littlewood maximal estimate, we have So, . With this γ, the above inequality becomes, Note that for the dyadic martingale { f n }, Recall the continuity property of Lebesgue measure, if {E n } is a sequence of sets with E n ⊂ E n+1 for all n and E = ∞ n=1 E n , then |E| = lim n→∞ |E n |. Using this we get, This completes the proof of the lemma. Precisely, it is unbounded a.e. on this set. Even though, one can obtain the rate of growth of | f n | on the set {x : S f (x) = ∞}. The rate of growth of | f n | can be obtained by the martingale analogue of Kolmogorov's law of the iterated logarithm. Stout [6] obtained the law of the iterated logarithm for dyadic martingales. Here we obtain the same upper bound in the law of the iterated logarithm for dyadic martingales using the estimates obtained in Lemma 3 and Borel-Cantelli Lemma (Lemma 1).