The Properties Of Random Power Series And Random Dirichlet Series

Random series was raised by Emile Broel in1896, but as a theory began in the twentieth century thirties because H. Steinhaus, REACPaley and A. Zygmund published several papers. Since then, in the30century and50’s to70’s, Chinese and foreign scholars, many studies were made on the random series, and achieved many important results.In recent years scholars studied random power series and Dirichlet series of con-vergence, growth and value distribution, has been a series of creative achievement, but the random factor in general are independent random variables. Although the assumption of independence at some point is reasonable, but to verify that a sample of independence but it is very difficult, and in some practical problems, samples are not independent of observations. The purpose of this article is the nature of random power series and the Dirichlet series on the plane which their coefficeient is is φ mixed sequence. First, I made a simple narrative about recent research findings. Secondly, give the φ mixed sequence of definitions, and proves that the φ mixed sequence of three series theorem and in two different conditions of two strong equivalence the-orem. Finaly,using strong law of large numbers study the growth of random power series under certain conditions and the convergence and growth of random Dirich-let series in the convergence half-plane when the coefficients is φ mixed sequence. Application of this article conclusions and methods, you can promote and improve a range of theorems, and allows the relevant issues of convenience and simplicity.