The Strong Convergence For Arbitrary Stochastic Sequences And The Strong Laws Of Large Numbers

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 and the strong laws of large numbers for the sequences of arbitrary r.v. on the certain partial sets . In this paper , firstly , the author studies the strong convergence on arbitrary stochastic sequences .By using martingale convergence theorem and stopping time ,the strong limit theorems are obtained on the certain local sets . Some classical strong laws of large numbers for sequences of independent r.v.'s and some convergence theorems for martingale difference sequences are the particular cases of the result . Secondly , the strong convergence of series on arbitrary stochastic sequences are studied . A strong limit theorem on thissequences 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 . Thirdly , the strong laws of large numbers for the *-mixing sequences is proved in different conditions .It is ease to obtain several classical strong laws . At the same time , the strong laws for the sequences of independent r.v., m-dependent r.v. and arbitrary stochastic sequences are discussed . Finally , the author constructs and testifies a sort of non-decreasing singular function by using the strong law of large numbers for nonhomogencus Markov chain . A strictly increasing singular function is obtained form the result . The method depends on the strong law of Markov chain ,rather than the transform of Cantor function.