Page 92 - Special Topic Session (STS) - Volume 3

STS517 Zaoli C. et al.
where is the vector with zero coordinates, the notation ≤ for vectors
= ( , . . . , ) and = ( , . . . , ) means that ≤ for all = 1, . . . , , and

1
1

the notation → ∞ means that all d components of the vector n tend to
infinity. Denote

What limit theorems does the array ( ) satisfy? It was shown by Leadbetter

and Rootzén (1998) that under appropriate strong mixing conditions, only the
classical three types of limiting distributions (Gumbel, Fréchet and Weibull)
may appear (even when forcing → ∞ along a monotone curve). In the case
when the marginal distributions of the field have regularly varying tails, this
allows only the Fréchet distribution as a limit.
In this paper we will discuss only random fields with regularly varying tails,
in which case the experience from the classical extreme value theory tells us
to look for limit theorems for the type

for some nondegenerate random variable . The regular variation of the
marginal distributions means that

(1.2) P (X (0) > x) = x L(x), α > 0, L slowly varying,
−α
see e.g. Resnick (1987). Notice that the assumption is only on the right tail of
the distribution since, in most cases, one does not expect a limit theorem for
the partial maxima as in (1.1) to be affected by the left tail of ().
If the random field consists of i.i.d. random variables satisfying the
regular variation condition (1.2), then the classical extreme value theory tells
us that the convergence in (1.1) holds if we choose

(1.3) = inf{ > 0: (() > ) ≤ ( … ) },
−1

1
in which case the limiting random variable has the standard Fréchet
distribution. We are interested in understanding how the spatial dependence
in the random field affects the scaling in and the distribution of the limit not
only in (1.1), but in its functional versions, which can be stated in different
spaces, for example in the space (ℝ ) of right continuous, with limits along

+
monotone paths, functions (see Straf (1972)), or in the space of random sup
measures ℳ(ℝ ); see O’Brien et al. (1990). We will describe the relevant

+
spaces below.
If the time is one-dimensional, and the memory in the stationary process
is short, then the standard normalization (1.3) is still the appropriate one, and
the limits both in (1.1) and its functional versions change only through a
change in a multiplicative constant; see Samorodnitsky (2016) and references
therein. However, when the memory becomes sufficiently long, both the order

81 | I S I W S C 2 0 1 9