Activity Rates with Very Heavy Tails

Activity Rates with Very Heavy Tail

Scaling Limits for Workload Process

Scaling Limits for Workload Proces

Large deviations for solutions to stochastic recurrence equations under Kesten's condition

In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten's condition [Acta Math. 131 (1973) 207-248] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise sequence of the equations can have light tails. The results of the paper are analogs to those obtained by A. V. Nagaev [Theory Probab. Appl. 14 (1969) 51-64; 193-208] and S. V. Nagaev [Ann. Probab. 7 (1979) 745-789] in the case of partial sums of i.i.d. random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies in the sequence. We apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model.Comment: Published in at http://dx.doi.org/10.1214/12-AOP782 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Auto-tail dependence coefficients for stationary solutions of linear stochastic recurrence equations and for GARCH(1,1)

We examine the auto-dependence structure of strictly stationary solutions of linear stochastic recurrence equations and of strictly stationary GARCH(1, 1) processes from the point of view of ordinary and generalized tail dependence coefficients. Since such processes can easily be of infinite variance, a substitute for the usual auto-correlation function is needed

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