Statistics & Probability Letters 65 (2003) 317–329 Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes Xavier Bardina a ; ∗; 1 , Maria Jolis a ;1 , Ciprian A. Tudor b a Departament de Matem atiques, Universitat Aut onoma de Barcelona, Bellaterra 08193, Barcelona, Spain b Laboratoire de Probabilit es; Universit e Paris 6; 4, Place Jussieu; Paris Cedex 05 75252, France Abstract We give a result of approximation in law, in the space of the continuous functions on [0; 1] 2 , of Gaussian two-parameter processes that can be represented in law as a certain Wiener-type integral. The approximations are constructed from a Poisson process in the plane. c  2003 Elsevier B.V. All rights reserved. Keywords: Fractional Brownian sheet; Weak convergence; Two-parameter Poisson process; Two-parameter Gaussian processes 1. Introduction We prove a result of approximation in law for Gaussian processes that we will denote by W K1;K2 = {W K1;K2 s;t ; (s;t ) ∈ [0; 1] 2 } such that W K1;K2 s;t :=  1 0  1 0 K 1 (s;u)K 2 (t;v)dB u;v ; (1) where B = {B s;t ; (s;t ) ∈ [0; 1] 2 } is a standard Brownian sheet and the deterministic kernels K 1 and K 2 satisfy certain conditions that we will specify in the section of preliminaries. The main example of this kind of processes is the fractional Brownian sheet (see Section 4.1). Observe that the processes given by (1) are characterized by the fact that they are centered, Gaussian and their covariance * Corresponding author. E-mail addresses: bardina@mat.uab.es (X. Bardina), mjolis@mat.uab.es (M. Jolis), tudor@ccr.jussieu.fr (C.A. Tudor). 1 Partially supported by DGES Grants BFM2000-0009, BFM2000-0607 and by CIRIT Grant 2001SGR00174. 0167-7152/$-see front matter c  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2003.09.001