Jon Aaronson

LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS

We introduce Gibbs–Markov maps T as maps with a (possibly countable) Markov partition and a certain type of bounded distortion property, and investigate its Frobenius–Perron operator P acting on (locally) Lipschitz continuous functions ϕ. If such a function ϕ belongs to the domain of attraction of a stable law of order in (0,2), we derive the expansion of the eigenvalue function t↦λ(t) of the characteristic function operators Ptf=Pfexp[i (perturbations of P) around 0. From this representation local and distributional limit theorems for partial sums ϕ+…+ϕ◦ Tn are easily obtained, provided ϕ is aperiodic. Applications to recurrence properties of group extensions are also given.

Ergodic theory for Markov fibred systems and parabolic rational maps

A paraboIic rational map of the Riemann sphere admits a non-atomic h-conformal measure on its Julia set where h = the Hausdorff dimension of the Julia set and satisfies 1/2 < h < 2. With respect to this measure the rational map is conservative, exact and there is an equivalent a-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework

Strong laws for L- and U-statistics

Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems; of Hoeffding and of Helmers for lid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.

Invariant measures and asymptotics for some skew products

For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures. These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic ergodic theorems, and bounded rational ergodicity.

The intrinsic normalising constants of transformations preserving infinite measures

We consider the collection of normalisations of a c.e.m.p.t. inside other c.e.m.p.t.s of which it is a factor. This forms an analytic, multiplicative subgroup ofR+. The groups corresponding to similar c.e.m.p.t.s coincide. “Usually” this group is {1}. Examples are given where the group is:R+, any countable subgroup ofR+, and also an uncountable subgroup ofR+ of any Haussdorff dimension. These latter groups are achieved by c.e.m.p.t.s which are not similar to their inverses.

Aperiodicity of cocycles and conditional local limit theorems

We establish conditions for aperiodicity of cocycles (in the sense of [12]), obtaining, via a study of perturbations of transfer operators, conditional local limit theorems and exactness of skew-products. Our results apply to a large class of Markov and non-Markov interval maps, including beta transformations. This allows us to establish various stochastic properties of beta expansions.

A Local Limit Theorem for Stationary Processes in the Domain of Attraction of a Normal Distribution

In this chapter, we prove local limit theorems for Gibbs-Markov processes in the domain of attraction of normal distributions.

Koksma’s Inequality and Group Extensions of Kronecker Transformations

We consider methods of establishing ergodicity of group extensions, proving that a class of cylinder flows are ergodic, coalescent and non-squashable. A new Koksma-type inequality is also obtained.

Second order ergodic theorems for ergodic transformations of infinite measure spaces

For certain pointwise dual ergodic transformations T we prove almost sure convergence of the log-averages ... (formule) and the Chung-Erdos averages ... (formule) towards ∫ f, where a(n) denotes the return sequence of T

On the structure of 1-dependent Markov chains

Any stationary 1-dependent Markov chain with up to four states is a 2-block factor of independent, identically distributed random variables. There is a stationary 1-dependent Markov chain with five states which is not, even though every 1-dependent renewal process is a 2-block factor.