13 research outputs found
A general correlation inequality and the Almost Sure Local Limit Theorem for random sequences in the domain of attraction of a stable law
In the present paper we obtain a new correlation inequality and use it for
the purpose of extending the theory of the Almost Sure Local Limit Theorem to
the case of lattice random sequences in the domain of attraction of a stable
law. In particular, we prove ASLLT in the case of the normal domain of
attraction of --stable law,
Edgeworth expansions in operator form
An operator form of asymptotic expansions for Markov chains is established.
Coefficients are given explicitly. Such expansions require a certain
modification of the classical spectral method. They prove to be extremely
useful within the context of large deviations.Comment: 12 page
On limit theorems for continued fractions
It is shown that for sums of functionals of digits in continued fraction
expansion the Kolmogorov-Feller weak laws of large numbers and the
Khinchine-L\'evy-Feller-Raikov characterization of the domain of attraction of
the normal law hold.Comment: 16 page
A limit theorem for random sums modulo 1
Residues of partial sums in a class of dependent random variables, including functionals of uniformly recurrent Markov chains, are in the domain of attraction of the uniform distribution. These types of limit theorems arise for example in the multiplication of floating-point numbers.
On moments of recurrence times for positive recurrent renewal sequences
The explicit formula for the moments of recurrence times for positive recurrent renewal sequences is established.
Edgeworth expansions in operator form
An operator form of asymptotic expansions for Markov chains is established. Coefficients are given explicitly. Such expansions require a certain modification of the classical spectral method. They prove to be extremely useful within the context of large deviations.
Almost sure local limit theorem for the Dickman distribution
We study the asymptotic behavior, and more precisely the second order properties, of the probabilistic model introduced in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002) for describing the Dickman distribution. This model appears as an extremal example in the theory of the local and almost sure local limit theorem. We establish a delicate correlation inequality for this system. We apply it to obtain a fine almost sure local limit theorem. In doing so, we also give a corrected proof of the corresponding local limit theorem stated in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002)