The bivariate negative binomial distribution is introduced using the Marshall-Olkin type bivariate geometrical distribution. It is used to the estimation of the distribution of the number of accidents in standard data.
For almost all infinite binary sequences of Bernoulli trials the frequency of blocks of length in the first terms tends asymptotically to the probability of the blocks, if increases like (for ) where tends to . This generalizes a result due to P. Flajolet, P. Kirschenhofer and R.F. Tichy concerning the case .
The commutator of multiplication by a function and a martingale transform of a certain type is a bounded operator on , , if and only if the function belongs to BMO. This is a martingale version of a result by Coifman, Rochberg and Weiss.
Boolean cluster point processes with various cluster distributions are examined by means of their spherical contact distribution function. Special attention is paid to clusters with strong independence properties (Poisson clusters) and regular clusters.
Passing from Cesàro means to Borel-type methods of summability we prove some ergodic theorem for operators (acting in a Banach space) with spectrum contained in ℂ∖(1,∞).