s for the PhD Course on Extremes in Space and Time, May 27-30 Extremes and sums of regularly varying observations Bojan Basrak (University of Zagreb) In the first part, we show how the dependence structure of extremes in a stationary regularly varying sequence can be described, using the concept of the tail process. This is illustrated on some standard time series models. In the second part, we...

Random vectors on the positive orthant whose distributions possess hidden regular variation are a subclass of those whose distributions are multivariate regularly varying. The concept is an elaboration of the coefficient of tail dependence of Ledford and Tawn(1996, 1997). We provide characterizations and examples of such distribution in terms of mixture models and product models.

For a sequence of random variables (X1, X2, . . . , Xn), n ≥ 1, that are independent and identically distributed with a regularly varying tail with index −α, α ≥ 0, we show that the contribution of the maximum term Mn , max(X1, . . . , Xn) in the sum Sn , X1 + · · · + Xn, as n → ∞, decreases monotonically with α in stochastic ordering sense.

We study the large-time asymptotic of renewal-reward processes with a heavy-tailed waiting time distribution. It is known that heavy tail distribution produces an extremely slow dynamics, resulting in singular large deviation function. When singularity takes place, bottom function flattened, manifesting anomalous fluctuations processes. In this article, we aim to how these singularities emerge ...

In this paper, we consider a one dimensional stochastic system described by an ellipticequation. A spatially varying random coefficient is introduced to account for uncertainty orimprecise measurements. We model the logarithm of this coefficient by a Gaussian process andprovide asymptotic approximations of the tail probabilities of the derivative of the solution.

Many interesting processes share the property of multivariate regular variation. This property is equivalent to existence of the tail process introduced by B. Basrak and J. Segers [1] to describe the asymptotic behavior for the extreme values of a regularly varying time series. We apply this theory to multivariate MA(∞) processes with random coefficients.

We investigate the maximal numberMk of offspring amongst all individuals in a critical Galton–Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index −α for 1 < α < 2, then k−1Mk converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

We consider a weighted stationary spherical Boolean model in R. Assuming that the radii of the balls in the Boolean model have regularly varying tails, we establish the asymptotic behaviour of the tail of the contact distribution of the thinned germ-grain model under 4 different thinning procedures of the original model.

Consider a stationary, pth order autoregression fX n g satisfying whose innovation sequence fZ n g is iid with regularly varying tail probabilities of index ?. From p of the autore-gressive coeecients and then to estimate the residuals by and then to apply Hill's estimator to the estimated residuals. We show that from the point of asymptotic variance, the second procedure is superior.

the objective followed in the present study was to survey the relationship between 18 body trait measurements (live weight, height at wither, paunch girth, neck diameter, body length, girth around the body, width of fat tail at above, below and midpoint of fat tail, fat tail length lowers right and left sides, fat tail gap length, fat tail depth at the above, below, and midpoint, and girth arou...