CENTRAL LIMIT THEOREM OF MIXED TYPE FOR TRANSFORMATIONS WITH QUASI-COMPACT PERRON-FROBENIUS OPERATORS

Takuya IKEDA, Takehiko MORITA
2017 Kyushu Journal of Mathematics  
We consider a nonsingular transformation whose Perron-Frobenius operator is quasi-compact on an appropriate Banach algebra. We establish the central limit theorem of mixed type with a nice convergence rate for a real-valued observable in the Banach algebra. As an application, we show that generalized piecewise expanding maps on the unit interval with Hölder continuous derivatives and the Banach algebra of Lebesgue integrable functions with a version of bounded p-variation satisfy our conditions
more » ... isfy our conditions if p is not smaller than the reciprocal of the Hölder exponent of the derivatives. where C (2) = C (0) C (1) /(1 − θ α ). Indeed, the inequality (5.4) is obvious. Next we have (5.5) as follows: where J ∈P n (A) denotes the summation taken over all J ∈ P n with T n J J = A. We first estimate the p-variation of L ng . By (5.6) and (5.9) we have It is easy to see that for f ∈ BV p and a subinterval J of [0, 1], we have where f J,∞ = sup x∈J |f (x)| and var p (f ; J ) is the total p-variation of the restriction f | J to J . Therefore, for J ∈ P n with T n J J = A, we have var p (χ A G n (T −n J )g(T −n J )) ≤
doi:10.2206/kyushujm.71.31 fatcat:7ajay2mnw5avjkep5ir532vbsm