Page 143 - Textos de Matemática Vol. 47

`=1
STONGLY DEPENDENT GAUSSIAN RANDOM FIELDS
133
q! \\pp!
P
{Mn,m(B`, 0)  1 ⇢n,m(un,m z ⇢n,m)} q x + zp2
⇠ P {Mn,m(B`, 0)  a + bn,m} `=1 ! n,m
\q Yq p = P {Nn⇤,m(B`) = 0} ! exp( ( (B`)e x +z
2 ),
n, m ! +1.
`=1 `=1
Finally, by dominated convergence and due to (3.2), we deduce
\q !\q !! P {Nn,m(B`) = 0} = P {Mn,m(B`)  un,m}
`=1
Z + 1 Yq ⇣ x + z p 2 ⌘ \q
! exp (B`)e (z)dz = P {N(B`) = 0} 1 `=1 `=1
`=1
as n, m ! +1.
As a corollary we obtain the convergence
P (Mn,m  un,m) ! Z +1 exp ⇣ e x +zp2 ⌘ (z)dz, 1
that generalizes the well known univariate result ([5]).
References
⇤
[1] R. Adler, The geometry of random field, John Wiley and Sons, New York, 1981.
[2] H. Choi, Central limit theory and extremes of random fields. PhD Thesis, Univ. of North
Carolina at Chapel Hill, 2002.
[3] J. Hu¨sler, Asymptotic approximation of crossing probabilities of random sequences, Z.
Wahrsch. Verw. Gebiete 63 (2), 257–270, 1983.
[4] O. Kallenberg, Random measures, Academic Press, London-New York, 1976.
[5] M. R. Leadbetter, G. Lindgren, and H. Rootz´en, Extremes and Related Properties of
Random Sequences and Processes, Springer-Verlag, Berlin, 1983.
[6] L. Pereira, On the extremal behavior of a nonstationary normal random field, J. Statist.
Plann. Inference 140 (11), 3567–3576, 2010.
[7] L. Pereira and H. Ferreira, Point processes of exceedances by random fields, J. Statist.
Plann. Inference 142 (3), 773–779, 2012.
[8] M. G. Temido, Mixture results for extremal behaviour of strongly dependent nonstatio-
nary Gaussian sequences, Test (9), 439–453, 2000.
(M. G. Temido) Department of Mathematics, University of Coimbra, Apartado 3008 EC Santa Cruz, 3001-501 Coimbra, Portugal
E-mail address: mgtm@mat.uc.pt
n, m ! +1,