scholarly journals Large deviation probabilities for sums of heavy-tailed dependent random vectors

1997 ◽  
Vol 13 (4) ◽  
pp. 647-660 ◽  
Author(s):  
Adam Jakubowski ◽  
Alexander. V. Nagaev ◽  
Zaigraev Alexander
Keyword(s):  
Large Deviation ◽  
Random Vectors ◽  
Large Deviation Probabilities ◽  
Heavy Tailed

scholarly journals Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

2015 ◽  
Vol 52 (1) ◽  
pp. 68-81 ◽  
Author(s):  
K. M. Kosiński ◽  
M. Mandjes
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Random Vectors ◽  
Deviation Principle ◽  
Logarithmic Asymptotics ◽  
Rate Functions ◽  
Regularly Varying ◽  
Positive Index

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

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