An almost sure invariance principle for the maps S_a(x)=ax(1-x) in the interval [0,1]
Abstract
In this paper we consider the family of maps in the unit interval defined by Sa(x)=ax(1-x), x∈[0,1], a∈[0,4]. For a countable set of values of a for which the a.c. Sa-invariant measure μaexists, we show an almost sure invariance principle for the process {f∘ Sia}, with f a function of bounded p-variation; from this the previously proved central limit theorem with respect to μa follows again; moreover log-log-laws and weak invariance principles for the process{f∘ Sia} follow.
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