Regularly Varying Measures on Metric Spaces: Hidden Regular Variation and Hidden Jumps

Stable random fields, point processes and large deviations

“…This results in the space M 0 (R) of all Borel measures on R\{0} that are finite outside any neighbourhood of 0. The convergence in M 0 (R) implies vague convergence in R\{0}; see Lemma 2.1. in Lindskog et al (2013).…”

“…Let M 0 (M q ) denote the space of all Borel measures ρ on M q satisfying ρ(M q \ B(Ø, ε)) < ∞ for all ε > 0 (here B(Ø, ε) is the open ball of radius ε around the null measure Ø in the vague metric). Define the Hult-Lindskog-Samorodnitsky (HLS) convergence ρ n → ρ in M 0 (M q ) by ρ n (f ) → ρ(f ) for all f ∈ C b,0 (M q ), the space of all bounded continuous functions on M q that vanish in a neighbourhood of Ø; see Theorem 2.1 in Hult and Lindskog (2006) and Theorem 2.1 in Lindskog et al (2013). This set up is the same as in Hult and Samorodnitsky (2010) except that the space M q includes all Radon measures in E q , not just the Radon point measures.…”

“…Their work relies on convergence of measures that was introduced in Hult and Lindskog (2006). See also the recent works of Das et al (2013) and Lindskog et al (2013), which extended this convergence to more general situations.…”