George L. O'Brien

The maximum term of uniformly mixing stationary processes

Let {Xn} be a uniformly (or strongly) mixing stationary process and let Zn=max(X1, X2,..., Xn). For ξ>0, let cn(ξ)=inf {xεR: n P(X1>x)≦ξ}. Under a condition which holds for all ϕ-mixing processes, necessary and sufficient conditions are given for P(Zn≦cn(ξ)) to converge to each possible limit. Some conditions for convergence of P(Zn≦dn) for any sequence dn are also obtained.

Sequences of capacities, with connections to large-deviation theory

Acapacity is a set function with some regularity properties on a Hausdorff spaceE. Many measures and all sup measures are examples. The set of capacities onE can be endowed with two natural topologies. The narrow topology corresponds to the weak topology for probability measures, while the vague topology corresponds to the vague topology for Radon measures. The connection between these topologies and large-deviation principles was noted in recent joint work with W. Vervaat. Here, the theory of capacities and their topologies is developed in directions which have implications for large-deviation theory.

Stationary self-similar extremal processes

SummaryLet (ξk)k∞=−∞ be a stationary sequence of random variables, and, forA⊂ℝ, let

Monotonicity of the number of self-avoiding walks

M_n (A): = \mathop V\limits_{k/n \in A} \gamma _n (\xi _k )

Relative Compactness for Capacities, Measures, Upper Semicontinuous Functions and Closed Sets

where γn is an affine transformation of ℝ (has the forman·+bn,an>0,bn∈ℝ). ThenMn is a random sup measure, that is,

The occurrence of large values in stationary sequences

M_n (\mathop U\limits_\alpha G_\alpha ) = \mathop V\limits_\alpha M_n (G_\alpha )