Abstract
In the paper we solve the limit problem for partial maxima of m-dependent stationary random fields and we extend the obtained solution to fields satisfying some local mixing conditions. New methods for describing the limitting distribution of maxima are proposed. A notion of a phantom distribution function for a random field is investigated. As an application, several original formulas for calculation of the extremal index are provided. Moving maxima and moving averages as well as Gaussian fields satisfying the Berman condition are considered.
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Appendix
Appendix
Two self-serving lemmas applied in the proof of Theorem 2.1 and in Section 2.2 are given below.
Lemma 7.1
LetN(n) →∞be such thatN∗(n)/nd →∞. Then there exists a sequence\(\{\boldsymbol {\psi }(n)\}\subset \mathbb {N}^{d}\)satisfyingψ(n) →∞, ψ1(n)ψ2(n)⋯ψd(n) ∼ nd, ψi(n) ≤ Ni(n) fori ∈{1, 2,…,d − 1} andψd(n)/Nd(n) → 0.
Proof
Let the sequences \(\{t_{i}(n)\}\subset \mathbb {R} \), for i ∈{1, 2,…,d − 1}, be chosen so that
and
for 2 ≤ i ≤ d − 1. Let us consider \({{a}}_{i}(n)\in \mathbb {R} \) defined as follows
Then, we easily get that ai(n) →∞ and ai(n) ≤ Ni(n) for i ∈{1, 2,…,d − 1}. We will show that also ad(n) →∞ and ad(n)/Nd(n) → 0. Indeed, by the definition of ad− 1(n), ad− 2(n), …, we have
Since td− 1(n) → ∞ and ad(n) ≥ td− 1(n), the condition ad(n) → ∞ holds. Moreover, applying nd = o(N∗(n)) and ti− 1(n) = o(Ni(n)Ni+ 1(n)⋯Nd(n)), we conclude that ad(n)/Nd(n) → 0. To complete, we shall define \(\psi _{i}(n):=\lfloor {{a}}_{i}(n) \rfloor \in \mathbb {N} \). □
Lemma 7.2
Fora(m, n) ≥ 0, \(b(m,n)\in \mathbb {R}\)(\(m\in \mathbb {N}_{+},n\in \mathbb {N}\)),\(b\in \mathbb {R}\),
-
(i)
if \(\limsup _{m\to \infty }\limsup _{n\to \infty }a(m,n) = 0\) holds, then \(\lim _{n\to \infty } a(m_{n},n) = 0\) ;
-
(ii)
if\(\lim _{m\to \infty }\lim _{n\to \infty } b(m,n) = b\)holds,then\(\lim _{n\to \infty }b(m_{n},n) = b\),
for some sequencern →∞and allmn →∞satisfyingmn ≤ rn.
Proof
To prove (i), observe that for every m and \(a(m):=\limsup _{n\to \infty } a(m,n)\), there exists \(N_{m}\in \mathbb {N}\) such that a(m, n) − a(m) ≤ 1/m for all n ≥ Nm. Let us choose Nm for each m so that the sequence {Nm} is increasing. Define . Then rn →∞ and for every mn →∞ satisfying mn ≤ rn we obtain
since \(n \geq N_{r_{n}} \geq N_{m_{n}}\) holds, and consequently
The proof of part (ii) is fully analogous. □
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Jakubowski, A., Soja-Kukieła, N. Managing local dependencies in asymptotic theory for maxima of stationary random fields. Extremes 22, 293–315 (2019). https://doi.org/10.1007/s10687-018-0336-6
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DOI: https://doi.org/10.1007/s10687-018-0336-6
Keywords
- Stationary random fields
- Extremes
- m-dependence
- Extremal index
- Phantom distribution function
- Moving averages
- Moving maxima
- Berman’s condition