39 research outputs found
On limit theorems for fields of martingale differences
We prove a central limit theorem for stationary multiple (random) fields of
martingale differences , ,
where is a action. In most cases the multiple
(random) fields of martingale differences is given by a completely commuting
filtration. A central limit theorem proving convergence to a normal law has
been known for Bernoulli random fields and in [V15] this result was extended to
random fields where one of generating transformations is ergodic. In the
present paper it is proved that a convergence takes place always and the limit
law is a mixture of normal laws. If the action is ergodic and , the limit law need not be normal. For proving the result mentioned above, a
generalisation of McLeish's CLT for arrays of martingale
differences is used. More precisely, sufficient conditions for a CLT are found
in the case when the sums converge only in distribution. The
CLT is followed by a weak invariance principle. It is shown that central limit
theorems and invariance principles using martingale approximation remain valid
in the non-ergodic case
Martingale-coboundary decomposition for stationary random fields
We prove a martingale-coboundary representation for random fields with a
completely commuting filtration. For random variables in L2 we present a
necessary and sufficient condition which is a generalization of Heyde's
condition for one dimensional processes from 1975. For Lp spaces with 2 \leq p
< \infty we give a necessary and sufficient condition which extends Volny's
result from 1993 to random fields and improves condition of El Machkouri and
Giraudo from 2016 (arXiv:1410.3062). In application, new weak invariance
principle and estimates of large deviations are found.Comment: Stochastics and Dynamics 201
Local limit theorem in deterministic systems
We show that for every ergodic and aperiodic probability preserving system,
there exists a valued, square integrable function such that
the partial sums process of the time series satisfies the lattice local limit theorem.Comment: 17 page
A strictly stationary -mixing process satisfying the central limit theorem but not the weak invariance principle
In 1983, N. Herrndorf proved that for a -mixing sequence satisfying the
central limit theorem and , the weak
invariance principle takes place. The question whether for strictly stationary
sequences with finite second moments and a weaker type (, ,
) of mixing the central limit theorem implies the weak invariance
principle remained open.
We construct a strictly stationary -mixing sequence with finite
moments of any order and linear variance for which the central limit theorem
takes place but not the weak invariance principle.Comment: 12 page
On the central and local limit theorem for martingale difference sequences
Let (\Omega, \A, \mu) be a Lebesgue space and an ergodic measure
preserving automorphism on with positive entropy. We show that there
is a bounded and strictly stationary martingale difference sequence defined on
with a common non-degenerate lattice distribution satisfying the
central limit theorem with an arbitrarily slow rate of convergence and not
satisfying the local limit theorem. A similar result is established for
martingale difference sequences with densities provided the entropy is
infinite. In addition, the martingale difference sequence may be chosen to be
strongly mixing.Comment: Accepte pour publication dans Stochastics and Dynamic
Stable CLT for deterministic systems
We show that for every ergodic and aperiodic probability preserving
transformation and there exists a function whose associated
time series is in the standard domain of attraction of a non-degenerate
symmetric -stable distribution.Comment: 17 pages, 0 figure
LOCAL LIMIT THEOREM IN DETERMINISTIC SYSTEMS
We show that for every ergodic and aperiodic probability preserving system, there exists a Z valued, square integrable function f such that the partial sums process of the time series f • T i ∞ i=0 satisfies the lattice local limit theorem