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Journal article
Convergence to type I distribution of the extremes of sequences defined by random difference equation
Stochastic processes and their applications, v 121(10), pp 2231-2242
2011
Abstract
We study the extremes of a sequence of random variables
(
R
n
)
defined by the recurrence
R
n
=
M
n
R
n
−
1
+
q
,
n
≥
1
, where
R
0
is arbitrary,
(
M
n
)
are iid copies of a non-degenerate random variable
M
,
0
≤
M
≤
1
, and
q
>
0
is a constant. We show that under mild and natural conditions on
M
the suitably normalized extremes of
(
R
n
)
converge in distribution to a double-exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence
(
R
n
)
under the assumption that
P
(
M
>
1
)
>
0
.
► We study extremes of sequences defined by random linear equations in the case where the limiting random variables have light tails. ► We show that under natural and mild conditions, properly normalized extremes converge to a double-exponential random variable. ► Our work partially complements earlier results by other researchers who studied convergence of extremes to a Fréchet distribution.
Metrics
Details
- Title
- Convergence to type I distribution of the extremes of sequences defined by random difference equation
- Creators
- Paweł Hitczenko - Drexel University
- Publication Details
- Stochastic processes and their applications, v 121(10), pp 2231-2242
- Publisher
- Elsevier
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000294592000003
- Scopus ID
- 2-s2.0-79960998136
- Other Identifier
- 991019168100104721
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Data related to this publication, from InCites Benchmarking & Analytics tool:
- Web of Science research areas
- Statistics & Probability