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Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings
Let W = { W n : n ∈ N } be a sequence of random vectors in R d , d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we find that log P (there exists n ∈ N : W n u q ) as u → ∞. We follow the approach of the restricted large d...
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| Published in: | Journal of applied probability 2015-03, Vol.52 (1), p.68-81 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | Let
W
= {
W
n
:
n
∈
N
} be a sequence of random vectors in
R
d
,
d
≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of
W
, that is, for any vector
q
> 0 in
R
d
, we find that log
P
(there exists
n
∈
N
:
W
n
u
q
) 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 {
a
n
}, {
v
n
}, (1 /
v
n
)log
P
(
W
n
/
a
n
≥
u
q
) has a, continuous in
q
, limit
J
W
(
q
). We allow the scalings {
a
n
} and {
v
n
} to be regularly varying with a positive index. This approach is general enough to incorporate sequences
W
, such that the probability law of
W
n
/
a
n
satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature. |
|---|---|
| ISSN: | 0021-9002 1475-6072 |
| DOI: | 10.1017/S0021900200012201 |