Page 93 - Special Topic Session (STS) - Volume 3

STS517 Zaoli C. et al.
of magnitude of the normalization in the limit theorems changes, and the
nature of the limit changes as well; see Samorodnitsky (2004) and Owada and
Samorodnitsky (2015b). Furthermore, the limit may even stop having the
Fréchet distribution (or Fréchet marginal distributions, in the functions limit
theorems); see Samorodnitsky and Wang (2017). It is reasonable to expect that
similar phenomena happen for random fields, but because it is harder to
quantify how long the memory is when the time is not one-dimensional, less
is known in this case.
In this paper we will concentrate on the case where the random field is
a symmetric -stable () random field, 0 < < 2. Recall that this means
that every finite linear combination of the values of the values of the random
field has a one-dimensional SαS distribution, i.e. has a characteristic function
of the form exp{− |θ| }, θ ∈ ℝ, where ∈ [0, ∞) is a scale parameter that

depends on the linear combination; see Samorodnitsky and Taqqu (1994). The
marginal distributions of random fields satisfy the regular variation
assumption (1.2) with 0 < < 2 that coincides with the index of stability. In
this case a series of results on the relation between the sizes of the extremes
of stationary SαS random fiels and certain ergodic-theoretical properties of
the Lévy measures of these fields is due to Parthanil Roy and his coworkers;
see Roy and Samorodnitsky (2008), Chakrabarty and Roy (2013), Sarkar and
Roy (2016). These results are made possible because of the connection
between the structure of the random fields and ergodic theory
established by Rosiński (2000).
This paper contributes to understanding the extremal limit theorems for
random fields and their connection to the dynamics of the Lévy measures.
In this sense our paper is related to the ideas of Rosiński (2000). However, we
will restrict ourselves to certain Markov flows. This will allow us to avoid, to a
large extent, the language of ergodic theory, and state everything in purely
probabilistic terms. There is not doubt, however, that our results could be
extended to more general dynamical systems acting on the Lévy measures of
random fields. The generality in which work is sufficient to demonstrate
the new phenomena that may arise in extremal limit theorems for random
fields with long range dependence. We will exhibit new types of limits, some
of which will have non-Fréchet distributions, both in the space of random sup
measures and in the space (ℝ ).

+

2. A random field with long range dependence
We start with a construction of a family of stationary SαS random fields,
0 < α < 2 , whose memory has a natural finite-dimensional
parameterization. It is an extension to random fields of models considered
before in the case of one-dimensional time; see e.g. Resnick et al. (2000),

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