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This paper considers the asymptotic distribution of the longest edge of the minimal spanning tree and nearest neighbor graph on X_1,...,X_{N_n} where X_1,X_2,... are i.i.d. in Re^2 with distribution F and N_n is independent of the X_i and satisfies N_n/nto_p1. A new approach based on spatial blocking and a locally orthogonal coordinate system is developed to treat cases for which F has unbounded support. The general results are applied to a number of special cases, including elliptically contoured distributions, distributions with independent Weibull-like margins and distributions with parallel level curves.
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Possible reasons for the uniqueness of the positive geometric law in the context of stability of random extremes are explored here culminating in a conjecture characterizing the geometric law. Our reasoning comes closer in justifying the geometric law in similar contexts discussed in Arnold et al. (1986) and Marshall & Olkin (1997) and also supplement their arguments.
We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the algebraic point of view of our recent work on Popa groups.
We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine whether the cutoff phenomenon occurs. We thoroughly discuss families of trees with cutoff, and show how our criteria can be used to prove the absence of cutoff for several classes of trees, including spherically symmetric trees, Galton-Watson trees of a fixed height, and sequences of random trees converging to the Brownian CRT.
The Maki-Thompson rumor model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals; namely, ignorants, spreaders and stiflers. A spreader tells the rumor to any of its nearest ignorant neighbors at rate one. At the same rate, a spreader becomes a stifler after a contact with other nearest neighbor spreaders, or stiflers. In this work we study the model on random trees. As usual we define a critical parameter of the model as the critical value around which the rumor either becomes extinct almost-surely or survives with positive probability. We analyze the existence of phase-transition regarding the survival of the rumor, and we obtain estimates for the mean range of the rumor. The applicability of our results is illustrated with examples on random trees generated from some well-known discrete distributions.
In this paper, joint asymptotics of powered maxima for a triangular array of bivariate powered Gaussian random vectors are considered. Under the Husler-Reiss condition, limiting distributions of powered maxima are derived. Furthermore, the second-order expansions of the joint distributions of powered maxima are established under the refined Husler-Reiss condition.
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