Consider a random vector $\textbf{U}$ whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $\textbf{U}$ , conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.

The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in Rd. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

Let {Xn, n ≥ 1} be an independent, identically distributed random sequence with each Xn having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.

In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

We propose a simple and efficient scheme for ranking all teams in a tournament where matches can be played simultaneously. We show that the distribution of the number of rounds of the proposed scheme can be derived using lattice path counting techniques used in ballot problems. We also discuss our method from the viewpoint of parallel sorting algorithms.

Let U1,U2,… be a sequence of i.i.d. random vectors distributed uniformly in a compact plane region A of unit area. Sufficient conditions on the geometry of A are provided under which the Euclidean diameter Dn of the first n of the points converges weakly upon suitable rescaling.

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X1,· ··, Xn whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

The time to extinction of a subcritical Galton–Watson branching process and the time of last mutation of its infinite-alleles version are maxima of independent random variables having an upper tail of geometric type, and hence they are not attracted to any extreme value distribution. It is shown that Anderson's asymptotic results for maxima of discrete variates are applicable, and this rectifies a false assertion made in respect to the infinite-alleles simple branching process.

We show that the simple characterizations given by Takahashi for the independence and the total dependence of a multivariate extreme value distribution do not hold for the larger class of maximum infinitely divisible (max i.d.) distributions. This holds also for sup self-decomposable distributions.

Let Yn denote the largest of n independent N(0,1) random variables. It is shown that the error in approximating the distribution of Yn by the type III extreme value distribution exp {– (–Ax + B)k}, k > 0, is uniformly of order (log n)–2 if and only if the constants A, B and k satisfy certain conditions. In particular, this holds for the penultimate form of Fisher and Tippett (1928). Furthermore, two sufficient conditions are given so that these results can be extended to a stationary Gaussian sequence.

Let Xnn denote the largest of n independent N(0, 1) variables. Several methods of estimating P(Xnn ≦ x) are considered. It is shown that X2nn, when normalized in an optimal way, converges to the extreme value distribution at a rate of only 1/(log n)2, and that if 0 < t ≠ 2 then |Xnn|t converges at a rate of 1/log n. Therefore it is not feasible to use the extreme value distribution to estimate probabilities for normal extremes unless the sample size is extremely large. An alternative approach is presented, which gives very good estimates of P(Xnn ≦ x) for n ≧ 10. The case of rth extremes is also considered.

Let Yn denote the largest of n independent N(0, 1) variables. It is shown that if the constants an and bn are chosen in an optimal way then the rate of convergence of (Yn – bn)/an to the extreme value distribution exp(–e–x), as measured by the supremum metric or the Lévy metric, is proportional to 1/log n.

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.