Nonconventional limit theorems

• 成果类型: Article
• 署名作者: Kifer, Yuri
• 署名单位: Hebrew University of Jerusalem
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-009-0223-9
• 发表日期: 2010
• 页码: 71-106
• 关键词: random transformations

摘要

The polynomial ergodic theorem ( PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337-349, 1987) and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1/N Sigma(N)(n=1) T-q1(n) f(1) ... T-ql(n) f(l) where T is a weakly mixing measure preserving transformation, f(i)'s are bounded measurable functions and q(i)'s are polynomials taking on integer values on the integers. Motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1/root N Sigma(N)(n=1)(F(X-0(n), X-1(q(1)(n)), X-2(q(2)(n)) ,..., X-l(q(l)(n))) - (F) over bar) where X-i's are exponentially fast psi - mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, (F) over bar = integral Fd(mu(0) x mu(1) x ... x mu(l)), mu(j) is the distribution of X-j (0), and q(i)'s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when qi's are polynomials of growing degrees. When F(x(0), x(1),..., x(l)) = x(0)x(1)x(2) ... x(l) exponentially fast alpha-mixing already suffices. This result can be applied in the case when X-i(n) = T-n f(i) where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as in the case when X-i (n) = f(i)(xi(n)) where xi(n) is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.