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Register a new userOn the Stability of Geometric Extremes
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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.
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In the context of stability of the extremes of a random variable X with respect to a positive integer valued random variable N we discuss the cases (i) X is exponential (ii) non-geometric laws for N (iii) identifying N for the stability of a given X and (iv) extending the notion to a discrete random variable X.
This article is concerned with the design and analysis of discrete time Feynman-Kac particle integration models with geometric interacting jump processes. We analyze two general types of model, corresponding to whether the reference process is in continuous or discrete time. For the former, we consider discrete generation particle models defined by arbitrarily fine time mesh approximations of the Feynman-Kac models with continuous time path integrals. For the latter, we assume that the discrete process is observed at integer times and we design new approximation models with geometric interacting jumps in terms of a sequence of intermediate time steps between the integers. In both situations, we provide non asymptotic bias and variance theorems w.r.t. the time step and the size of the system, yielding what appear to be the first results of this type for this class of Feynman-Kac particle integration models. We also discuss uniform convergence estimates w.r.t. the time horizon. Our approach is based on an original semigroup analysis with first order decompositions of the fluctuation errors.
We propose a geometric approach for bounding average stopping times for stopped random walks in discrete and continuous time. We consider stopping times in the hyperspace of time indexes and stochastic processes. Our techniques relies on exploring geometric properties of continuity or stopping regions. Especially, we make use of the concepts of convex sets and supporting hyperplane. Explicit formulae and efficiently computable bounds are obtained for average stopping times. Our techniques can be applied to bound average stopping times involving random vectors, nonlinear stopping boundary, and constraints of time indexes. Moreover, we establish a stochastic characteristic of convex sets and generalize Jensens inequality, Walds equations and Lordens inequality, which are useful for investigating average stopping times.
For extreme value copulas with a known upper tail dependence coefficient we find pointwise upper and lower bounds, which are used to establish upper and lower bounds of the Spearman and Kendall correlation coefficients. We shown that in all cases the lower bounds are attained on Marshall--Olkin copulas, and the upper ones, on copulas with piecewise linear dependence functions.
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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