Spectral criteria, SLLN's and as convergence of series of stationary variables

• 成果类型: Article
• 署名作者: Houdre, C; Lacey, MT
• 署名单位: Indiana University System; Indiana University Bloomington
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1996
• 页码: 838-856
• 关键词: stable processes; ergodic averages

摘要

It is shown here how to extend the spectral characterization of the strong law of large numbers for weakly stationary processes to certain singular averages. For instance, letting {X(t), t is an element of R(3)} be a weakly stationary field, {X(t)} satisfies the usual SLLN (by averaging over balls) if and only if the averages of {X(t)} over spheres of increasing radii converge pointwise. The same result in two dimensions is false. This spectral approach also provides a necessary and sufficient condition for the a.s. convergence of some series of stationary variables.