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We show that the class of lattice convolution equivalent distributions is not closed under convolution roots. We prove that the class of lattice convolution equivalent distributions is closed under convolution roots under the assumption of the exponentially asymptotic decreasing condition. This result is extended to the class $$\mathcal {S}_{\Delta }$$ of $$\Delta $$ -subexponential distributions. As a corollary, we show that the class $$\mathcal {S}_{\Delta }$$ is closed under convolution roots in the class $$\mathcal {L}_{\Delta }$$ . Moreover, we prove that the class of lattice convolution equivalent distributions is not closed under convolutions. Finally, we give a survey on the closure under convolution roots of the other distribution classes.

We investigate the asymptotic behaviour of the difference between the tails of a self-decomposable distribution with a two-sided regularly varying density on the real line and its Levy measure. Moreover, we study the second-order asymptotic behaviour of the tail of the t-th convolution power of a self-decomposable distribution with a two-sided regularly varying density.

First the relation between shift self-similar additive sequences and stationary sequences of Ornstein–Uhlenbeck type (OU type) on $$\mathbb {R}^d$$ is shown, and then the rates of escape for shift self-similar additive sequences are discussed. As a corollary, fundamental problems on recurrence of stationary sequences of OU type are solved. Some applications to laws of the iterated logarithm for strictly stable Levy processes on $$\mathbb {R}^d$$ and independent Brownian motions are given.

The class of exponential tilts of convolution equivalent distributions is determined. As a corollary, the local subexponentiality of one-sided infinitely divisible distributions is characterized. It is applied to the subexponentiality of the densities of a self-decomposable distribution and its Levy measure. Bondesson’s conjecture on the density of the Levy measure of a lognormal distribution is solved as an example. Results of Denisov et al. on the distributions of random sums are extended to the two-sided case. Finally, the local subexponentiality of the distribution of the supremum of a random walk is characterized.

Natural examples of increasing shift self-similar additive random sequences are constructed, which are associated with supercritical branching processes. The rate of growth and the distributional properties of them are studied in terms of the offspring distributions of the supercritical branching processes. The results are applied to two types of laws of the iterated logarithm for a Brownian motion on the unbounded Sierpinski gasket. An extension of the Bingham–Doney–de Meyer theorem on the limits of supercritical branching processes is also proved.

Absolute continuity and smoothness of distributions in the nested subclasses ~Lm(B), m = 0, 1, 2,..., of the class of all B-decomposable distributions are studied. All invertible matrices are classified into two types in terms of P.V. numbers. The minimum integer m for which all full distributions in ~Lm(B) are absolutely continuous and the minimum integer m for which all absolutely continuous distributions in ~Lm(B) have the densities of class Cr for 0 ≤ r ≤ ∞ are discussed according to the type of the matrix B related to P.V. numbers.