首页 | 本学科首页   官方微博 | 高级检索  
     检索      

随机狄里克莱级数的一些性质
引用本文:余家荣.随机狄里克莱级数的一些性质[J].数学学报,1978,21(2):97-118.
作者姓名:余家荣
作者单位:武汉大学
摘    要:本文研究随机狄里克莱级数的a.s.(几乎必然)收敛性和在a.s.收敛半平面内的a.s.增长性;为此,还研究了狄里克莱级数在收敛半平面内的增长性.这里推进了Valiron G.和Arnold L.的有关结果.文中还证明了在一定条件下,两类随机狄里克莱级数a.s.以其收敛轴上每一点为其Picard点或Borel(R)点.

收稿时间:1977-1-31

SOME PROPERTIES OF RANDOM DIRICHLET SERIES
Institution:YU JIA-RONG(Wuhan University)
Abstract:1.Consider an analytic function (s) defined by a Dirichlet series whose axis of convergence is σ = 0, where 0 ≤λ_n< + ∞ and lim (log n/λ_n) = 0. We define the Ritt order and proximate order of f(s) in σ > 0 and have obtained some relations between the growth of f(s) and the coefficients, which extend some of G. Valiron's results.2. Let (Ω, P) be a probability space. Consider a random Dirichlet series whose abscissa of convergence is σ(ω), where a_n(ω) are random variables in (Ω,P) and λ_n satisfy the same conditions as in 1(n = 0,1,2,…). To σ(ω) we extend L. Arnold's results on the radius of a.s. convergence of random Taylor series and have solved some of his problems.In the case σ(ω) = 0 a.s. the results mentioned in 1 is applied to studying the a.s. growth of f(s, ω) in σ> 0. As a special case we find that if |a_n(ω)| are independent and have the same non-degenerate distribution function F(x) and if the radius of a.s. convergence of g(s, ω)is 1, then the a.s. growth of g(s, ω |z|<1 ean be determined by the convergence or divergence of(logx)~k dF(x) (k≥1).3. Let the probability space in 2 be such that Ω = 0, 1], is composed of all Lebesgue measurable sets E on Ω and PE] is the Lebesgue measure of E. Let a_n(ω)= b_nε_n(ω) or b_nγ_n(ω), where {ε_n(ω)} or {γ_n(ω)}is a Rademaeher or Steinhaus sequenee and b_n are constants such that the abscissa of convergence of f(s, ω) is zero. Then, under certain conditions every point on σ = 0 is a Pieard or Borel (R) point of f(s, ω) a.S.
Keywords:
本文献已被 CNKI 等数据库收录!
点击此处可从《数学学报》浏览原始摘要信息
点击此处可从《数学学报》下载免费的PDF全文
设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号