Heavy Precise Asymptotics In The Law Of Large Numbers

This thesis is finished during my Master of Science and it consists of two chapters.Since Heyde proved the precise asymptotics for Hsu-Robbins-Erdos's law of large numbers in 1975, the compact and intuitionistic result have drawn many attentions from scholars of probability theory. Sp?taru proved that the precise asymptotics in Spitzer's law of large numbers. Gut and Sp?ataru generalized Spataru's result. Moreover they have proved more general result ?the precise asymptotics in Baum-Katz's law of large numbers in the same paper. With the considerable improvement in the study of law of large numbers in multidimensionally indexed random variables sequence, many law's precise asymptotics have been studied.H黶ler and Klesov generalized Heydel's result.Gut and Spatarul proved the precise asymptotics of Baum-Katz's law of large numbers in multidimensionally indexed case.In the first chapter , we discuss the precise asymptotics in LIL and Davis's law of large numbers for self-normalized sums. Gut and Sp?taru discussed the precise asymptotics in LIL for i.i.d random variables sums with the variance exist. Gut and Sp?taru also discussed the precise asymptotics in Davis's law of large numbers for i.i.d random variables sums. In this paper we generalized above results to self-normalized sums's case,i.e.Where TV denote the standard normal random variable .In the second chapter we discuss the precise asymptotics in LIL for multidimensionally indexed case. Gut and Spotarut discussed one of the precise asymptotics in LIL for multidimensionally indexed case under the condition E[X²(log(l + |X|))^d-1(loglog(e + ∞. In this paper , we discuss the other two precise asymptotics,i.e.Theorem 2.2.1 Let EX = 0, EX2 = a2 and for S > 0, E[X2(\og\og\X\)1+s] < oo, then we haveTheorem 2.2.3 Let EX - 0,and EX2 = a2 < oo, then we have...