In this paper, we first discuss the history of the law of the iterated logarithm. We then focus our discussion on how it was introduced in analysis. Finally we mention different types of law of the iterated logarithm and state some of the recent developments. In order to discuss the history and developments of law of the iterated logarithm, some definitions and theorems are in order: Definition...

By using Lebesgue bounded convergence theorem, we prove precise asymptotics in the law of iterated logarithm for independent and identically distributed random variables under sublinear expectation.

The stochastic properties of p-random sequences are studied in this paper. It is shown that the law of the iterated logarithm holds for p-random sequences. This law gives a quantitative characterization of the density of p-random sets. When combined with the invari-ance property of p-random sequences, this law is also useful in proving that some complexity classes have p-measure 0.

In this paper, we obtain the central limit theorem and the law of the iterated logarithm for Galton-Watson branching processes with time-dependent immigration in varying environments.

We find conditions which are sufficient and nearly necessary for the compact and bounded law of the iterated logarithm for Markov chains with a countable state space.

SUMMARY. For a stationary *-mixing stochastic process, the law of iterated logarithm, asymptotic normality and weak convergence to Brownian motion processes