Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$ , where $a_n, b_n\in \{-1,1\}$ . The associated Borel probability measure

$$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$

$\mu _{M, \{D_n\}}$

$3\mid \rho _i$

$i=1, 2$

$a_n=b_n=1$

$n\in {\mathbb N}$

In this paper we study a transform introduced by De Pril (1989) for recursive evaluation of convolutions of counting distributions with a positive probability in zero. We discuss some cases where the evaluation of this transform is simplified and relate the transform to infinitely divisible distributions. Finally we discuss an algorithm presented by Dhaene & Vandebroek (1994) for recursive evaluation of convolutions.

When does a distribution F have the property of both being in the domain of attraction of exp {–e–x} and having a second convolution-power tail equivalent to the first: Sufficient conditions and examples are given.

Majorisation is used to compare continuous distributions in terms of randomness. General results on randomness in the continuous case are given and these are used to investigate the connection between randomness and parameter values in some well-known families of distributions including the normal and gamma.

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.