Point processes associated with stationary stable processes

Point processes associated with stationary stable processe

Ruin Probability with Certain Stationary Stable Claims Generated by Conservative Flows

Ruin Probability with Certain Stationary Stable Claims Generated by Conservative Flow

Asymptotic Analysis of Exceedance Probability with Stationary Stable Steps

Asymptotic Analysis of Exceedance Probability with Stationary Stable Step

Scaling Limits for Workload Process

Scaling Limits for Workload Proces

Group theoretic dimension of stationary symmetric \alpha-stable random fields

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (2004a,b), where it was established, based on the seminal works of Rosi\'nski (1995,2000), that the growth rate is connected to the ergodic theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by Z^d in Roy and Samorodnitsky (2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to the fields indexed by R^d.Comment: To appear in Journal of Theoretical Probability. Affiliation of the authors are update

Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

We find a general formula for the distribution of time-averaged observables for systems modeled according to the sub-diffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly non-ergodic statistical mechanical framework is constructed, which is based on L\'evy's generalized central limit theorem. As an example we calculate the distribution of $\bar{X}$: the time average of the position of the particle, for unbiased and uniformly biased particles, and show that $\bar{X}$ exhibits large fluctuations compared with the ensemble average $$.Comment: 5 pages, 2 figure

Modeling and Analysis of Uncertain Time-Critical Tasking Problems (UTCTP)

Modeling and Analysis of Uncertain Time-Critical Tasking Problems (UTCTP

Weak Convergence of the function-indexed integrated periodogram for infinite variance processes

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $\alpha$-stable innovations. Under suitable summability conditions on the series of the Fourier coefficients of the index functions, we show that the weak limits constitute $\alpha$-stable processes which have representations as infinite Fourier series with i.i.d. $\alpha$-stable coefficients. The cases $\alpha\in(0,1)$ and $\alpha\in[1,2)$ are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case $\alpha\in(0,1)$, entropy conditions are needed for $\alpha\in[1,2)$ to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ253 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm