Summary
Letf be a real-valued function with period 1 satisfying some regularity conditions. It will be proved that, for anyθ 1,...,θ d > 1, the distribution of\(n^{ - 1/2} \sum\nolimits_{k = 1}^n {\left( {f(\theta _1^k t),...,f(\theta _d^k t)} \right)} \) on the probability space ([0, 1],dt) converges to a normal distribution whose covariance is given by algebraic relations among theθ i 's. This generalizes the classical work by M. Kac and refines the characterization off to have a degenerate limit. It also shows that the limit law of\(\sum\nolimits_{k = 1}^n {f(\theta _1^k t)/\sum\nolimits_{l = 1}^n {f(\theta _j^l t)} } \) is in most cases a Cauchy distribution.
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Fukuyama, K. The central limit theorem for Riesz-Raikov sums. Probab. Th. Rel. Fields 100, 57–75 (1994). https://doi.org/10.1007/BF01204953
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DOI: https://doi.org/10.1007/BF01204953
Mathematics Subject Classification (1991)
- 60F05
- 42A55
- 11K70