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The study completes a series of the author’s works devoted to the spread of particles population in supercritical catalytic branching random walk (CBRW) on a multidimensional lattice. The CBRW model describes the evolution of a system of particles combining their random movement with branching (reproduction and death) which only occurs at fixed points of the lattice. The set of such catalytic points is assumed to be finite and arbitrary. In the supercritical regime the size of population, initiated by a parent particle, increases exponentially with positive probability. The rate of the spread depends essentially on the distribution tails of the random walk jump. If the jump distribution has “light tails”, the “population front”, formed by the particles most distant from the origin, moves linearly in time and the limiting shape of the front is a convex surface. When the random walk jump has independent coordinates with a semiexponential distribution, the population spreads with a power rate in time and the limiting shape of the front is a star-shape nonconvex surface. So far, for regularly varying tails (“heavy” tails), we have considered the problem of scaled front propagation assuming independence of components of the random walk jump. Now, without this hypothesis, we examine an “isotropic” case, when the rate of decay of the jumps distribution in different directions is given by the same regularly varying function. We specify the probability that, for time going to infinity, the limiting random set formed by appropriately scaled positions of population particles belongs to a set $B$ containing the origin with its neighborhood, in $\mathbb{R}^d$. In contrast to the previous results, the random cloud of particles with normalized positions in the time limit will not concentrate on coordinate axes with probability one.

Financial mathematics is one of the most natural applications for the statistical analysis of time series. Financial time series reflect simultaneous activity of a large number of different economic agents. Consequently, one expects that methods of statistical physics and the theory of random processes can be applied to them.

In this paper, we provide a statistical analysis of time series of the FOREX currency market. Of particular interest is the comparison of the time series behavior depending on the way time is measured: physical time versus trading time measured in the number of elementary price changes (ticks). The experimentally observed statistics of the time series under consideration (euro–dollar for the first half of 2007 and for 2009 and British pound – dollar for 2007) radically differs depending on the choice of the method of time measurement. When measuring time in ticks, the distribution of price increments can be well described by the normal distribution already on a scale of the order of ten ticks. At the same time, when price increments are measured in real physical time, the distribution of increments continues to differ radically from the normal up to scales of the order of minutes and even hours.

To explain this phenomenon, we investigate the statistical properties of elementary increments in price and time. In particular, we show that the distribution of time between ticks for all three time series has a long (1-2 orders of magnitude) power-law tails with exponential cutoff at large times. We obtained approximate expressions for the distributions of waiting times for all three cases. Other statistical characteristics of the time series (the distribution of elementary price changes, pair correlation functions for price increments and for waiting times) demonstrate fairly simple behavior. Thus, it is the anomalously wide distribution of the waiting times that plays the most important role in the deviation of the distribution of increments from the normal. As a result, we discuss the possibility of applying a continuous time random walk (CTRW) model to describe the FOREX time series.