Addendum to: An iterated logarithm law for families of Brownian paths

The Law of Iterated Logarithm for the Square Integral of Brownian Motion

The paper aims to present recent developments about the inference of univariate distributions based on record values. Estimation of parameters, test of hypothesis, characterization problems and recurrence relations of moment and entropy of univariate distributions are discussed. Let X(t) R t 0 W(s) 2 ds where W is Brownian motion. Cameron and Martin (1944) found the distribution of X(t). Strass...

The Law of the Iterated Logarithm for Algorithmically Random Brownian Motion

Algorithmic randomness is most often studied in the setting of the fair-coin measure on the Cantor space, or equivalently Lebesgue measure on the unit interval. It has also been considered for the Wiener measure on the space of continuous functions. Answering a question of Fouché, we show that Khintchine’s law of the iterated logarithm holds at almost all points for each Martin-Löf random path ...

On the law of the iterated logarithm.

The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution...

An Analogue of the Law of the Iterated Logarithm

Laws of the iterated logarithm for α-time Brownian motion

We introduce a class of iterated processes called α-time Brownian motion for 0 < α ≤ 2. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric α-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in [14] for iterated Brownian motion. When α = 1 it takes the following form l...