This work is a continuation of [5], in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. Our main finding here is that most of the particles in an extreme level set come from only a small fraction of the clusters, which are atypically large.
This paper presents an empirical investigation of the intraday Brazilian stock market price fluctuations, considering q-Gaussian distributions that emerge from a nonextensive statistical mechanics. Our results show that, when returns are measured over intervals less than one hour, the empirical distributions are well fitted by q-Gaussians with exponential damped tails. Scaling behavior is also observed for these microscopic time intervals. We find that the time evolution of the distributions is according to a super diffusive q-Gaussian stationary process within a nonlinear Fokker-Planck equation. This regime breaks down due to the exponential fall-off of the tails, which in turn, governs the transient dynamics to the long-term macroscopic Gaussian regime. Our results suggest that this modeling provides a framework for the description of the dynamics of stock markets intraday price fluctuations.
We investigate the statistics of volumes of shares traded in stock markets. We show that the stochastic process of trading volumes can be understood on the basis of a mixed Poisson process at the microscopic time level. The beta distribution of the second kind (also known as q-gamma distribution), that has been proposed to describe empirical volume histograms, naturally results from our analysis. In particular, the shape of the distribution at small volumes is governed by the degree of granularity in the trading process, while the exponent controlling the tail is a measure of the inhomogeneities in market activity. Furthermore, the present case furnishes empirical evidence of how power law probability distributions can arise as a consequence of a fluctuating intrinsic parameter.
We consider an exactly solvable model of branching random walk with random selection, which describes the evolution of a population with N individuals on the real line. At each time step, every individual reproduces independently, and its offspring are positioned around its current locations. Among all children, N individuals are sampled at random without replacement to form the next generation, such that an individual at position x is chosen with probability proportional to e βx . We compute the asymptotic speed and the genealogical behavior of the system.
We consider a continuous time random walk on the rooted binary tree of depth n with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by 2 n+1 n and then centered by (log 2)n − log n, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree.
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