The Strong Convergence For Arbitrary Stochastic Sequences

Probability theory is a branch of mathematics dealing with chance phenomena and has clearly discernible links with the real world . It is the framework foundations of many applying subjects , such as Information theory , Mathematics Risk theory and Insurance theory for Actuaries etc . The strong limit theorems for partial sums of random variables (r.v.) is one of the central question for studying probability . Martingales and stopping times are the basis of Finance theory . Ruin theory ,Risk theory and Insurance theory . It is important meaningful to study the strong limit theorems for the sequences of r.v. by using martingale and stopping times.The purpose of this thesis is to study the strong convergence for the sequences of arbitrary r.v. on the certain partial sets . In this paper , firstly , the author studies the strong limit theorems on arbitrary stochastic sequences. A strong limit theorem on this sequences is proved on the certain partial set by using the convergence theorem of martingale difference and the property of conditional expectation and stopping time. As corollaries , some strong limit theorems of martingale difference sequences and a class of strong limit theorems of equitable ratios for an arbitrary stochastic sequences, a class of strong limit theorems for Markov process are obtained. Secondly , the three-series theorem are studied. By using the convergence theorem of martingale and the property of conditional expectation . As corollaries , It is ease to obtain several classical three-series theorems. Finally , the strong convergence of series on arbitrary stochastic sequences are studied . A strong limit theorem on this sequences is proved on the certain partial set by using the convergence theorem of martingale difference and the property of conditional expectation . As corollaries , some strong limit theorems of martingale difference sequences and the sequences of independent r.v. are obtained.