We develop a framework for regularly varying measures on complete separable metric spaces S with a closed cone C removed, extending material in [14,23]. Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in R ∞ + with marginal distributions having regularly varying tails and to càdlàg Lévy processes whose Lévy measures have regularly varying tails. In both cases, an infinite number of regular variation properties coexist distinguished by different scaling functions and state spaces.2010 Mathematics Subject Classification. 28A33,60G17,60G51,60G70.

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We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

We look at joint regular variation properties of MA(∞) processes of the form X = (X k , k ∈ Z) where X k = ∞ j=0 ψ j Z k−j and the sequence of random variables (Z i , i ∈ Z) are i.i.d. with regularly varying tails. We use the setup of M O -convergence and obtain hidden regular variation properties for X under suitable summabality conditions on the constant coefficents (ψ j : j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

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