arXiv subjects

Istvan Berkes

Publications and source records attributed to Istvan Berkes.

23 records · Page 2Linked to original sources

arXiv2012

On convergence and growth of sums $\sum c_k f(kx)$

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^γ<\infty$, $γ>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ for any increasing sequence $(n_k)$ of integers. We also construct an example showing that the previous condition is not sufficient for $γ<2$. Finally we give an a.e. bound for the growth of sums $\sum_{k=1}^N f(n_kx)$ differing from the corresponding optimal result for trigonometric sums by a loglog factor.

math.CA↗

arXiv2012

On series of dilated functions

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number theoretic factors, but precise results exist only in a few special cases. In this paper we use connections with orthogonal function theory and GCD sums to prove several new results, improve old ones and also to simplify and unify the theory.

math.CA↗

arXiv2009

A general strong law of large numbers for additive arithmetic functions

Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables, then $$ \lim_{N\to \infty} {\sum_{n=1}^N f(n) X_n \over\sum_{n=1}^N f(n)} \buildrel{a.s.}\over{=} \E X . $$

math.NT↗

arXiv2009

Estimation in nonstationary random coefficient autoregressive models

We investigate the estimation of parameters in the random coefficient autoregressive model. We consider a nonstationary RCA process and show that the innovation variance parameter cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for the remaining model parameters is proven so the unit root problem does not exist in the random coefficient autoregressive model.

stat.ME↗

arXiv2005

Near-integrated GARCH sequences

Motivated by regularities observed in time series of returns on speculative assets, we develop an asymptotic theory of GARCH(1,1) processes {y_k} defined by the equations y_k=σ_kε_k, σ_k^2=ω+αy_{k-1}^2+βσ_{k-1}^2 for which the sum α+βapproaches unity as the number of available observations tends to infinity. We call such sequences near-integrated. We show that the asymptotic behavior of near-integrated GARCH(1,1) processes critically depends on the sign of γ:=α+β-1. We find assumptions under which the solutions exhibit increasing oscillations and show that these oscillations grow approximately like a power function if γ\leq 0 and exponentially if γ>0. We establish an additive representation for the near-integrated GARCH(1,1) processes which is more convenient to use than the traditional multiplicative Volterra series expansion.

math.PR↗