We extend the classical Hsu-Robbins-Erdos theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log+X)[alpha]}<[infinity] for some [alpha]>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that...</[infinity]>

We introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data-based recovery of the process from repeated observations. A study of its asymptotic properties, as the frequency...

Consider an i.i.d. random field {Xk:k∈Z+d}, together with a sequence of unboundedly increasing nested sets Sj=⋃k=1jHk,j≥1, where the sets Hj are disjoint. The canonical example consists of the hyperbolas Hj={k∈Z+d:|k|=j}. We are interested in the number of “hyperbolas” Hj that...

The classical Marcinkiewicz-Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469-482] to random fields. Therein all indices...