This was first proved for Bernoulli random variables by Khintchine. Salem and Zygmund [SZ2] considered the case when the Xk are replaced by functions ak cosnkx on [−π, π] and gave an upper bound ( ≤ 1) result; this was extended to the full upper and lower bound by Erdös and Gál [EG]. Takahashi [T1] extends the result of Salem and Zygmund: Consider a real measurable function f satisfying f(x + 1...

Let X, X1, X2, . . . be i.i.d. random variables with mean zero and positive, finite variance σ2, and set Sn = X1 + . . . + Xn, n ≥ 1. We prove that, if EX2I{|X| ≥ t} = o((log log t)−1) as t →∞, then for any a > −1 and b > −1, lim 2↗1/√1+a ( 1 √ 1+a − 2)b+1 ∞n=1 (log n) a(log log n)b n P { maxk≤n |Sk| ≤ √ σ2π2n 8 log log n(2 + an) } = 4 π ( 1 2(1+a)3/2 )b+1Γ(b + 1), whenever an = o(1/ log log n).

In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norm-ing in the spirit of the classical Law of Iterated Logarithm.

Let X,X1, X2, . . . be a sequence of centered iid random variables. Let f(n) be a strongly additive arithmetic function such that ∑ p<n f2(p) p → ∞ and put An = ∑ p<n f(p) p . If EX2 < ∞ and f satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm: lim sup N→∞ ∑N n=1 f(n)Xn AN √ 2N log logN a.s. = ‖X‖2. We also prove the validity of the corresponding weighted...

A small ball problem and Chung’s law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit established.

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0) > 0, f'(t0) < 0, and f' is continuous in a neighborhood of t0, then [Formula: see text]almost surely where [Formula: see text]here [Formula: see text] is the two-sided Strassen limit set on [Formula: see text]. The proof relies on laws of the iterated logarith...