Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hilde...Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of Sigma (infinity)(n=1) X-n under a hypothesis that Sigma (infinity)(n=1) rho (X-n, c(n)) = infinity whener Sigma (infinity)(n=1) c(n) diverges, where the notation rho (X,c) denotes the Levy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of F-n(x(0)) with the limit value 0 < L-0 < 1, together with the essential convergence of Sigma (infinity)(n=1) X-n, is both sufficient and necessary in order for the series Sigma (infinity)(n=1) X-n to a.s. coverage. Moreover, if the essential convergence of Sigma (infinity)(n=1) X-n is strengthened to limsup(n=infinity) P(\S-n\ < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of Sigma (infinity)(n=1) X-n. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand([1]) as well.展开更多
文摘Let Sigma (infinity)(n=1) X-n be a series of independent random variables with at least one non-degenerate X-n, and let F-n be the distribution function of its partial sums S-n = Sigma (n)(k=1) X-k. Motivated by Hildebrand's work in [1], the authors investigate the a.s. convergence of Sigma (infinity)(n=1) X-n under a hypothesis that Sigma (infinity)(n=1) rho (X-n, c(n)) = infinity whener Sigma (infinity)(n=1) c(n) diverges, where the notation rho (X,c) denotes the Levy distance between the random variable X and the constant c. The principal result of this paper shows that the hypothesis is the condition under which the convergence of F-n(x(0)) with the limit value 0 < L-0 < 1, together with the essential convergence of Sigma (infinity)(n=1) X-n, is both sufficient and necessary in order for the series Sigma (infinity)(n=1) X-n to a.s. coverage. Moreover, if the essential convergence of Sigma (infinity)(n=1) X-n is strengthened to limsup(n=infinity) P(\S-n\ < K) = 1 for some K > 0, the hypothesis is already equivalent to the a.s. convergence of Sigma (infinity)(n=1) X-n. Here they have not only founded a very general limit theorem, but improved the related result in Hildebrand([1]) as well.
