Let X (t) be a random fun ction d erived , in a sense t hat is explained in t he paper , from a Poisson pTocess . It is proved t hat , under certain a ss umption s, t he dist rib ut ion of the r' +T . fUll ctional J, V(u, X(U» dlt ten ds to a Laplacian (i. e. norma l) distribu tion as T-->oo . This result is extended to t he case t hat X (t) is itself a Laplacian process by m eans of a t heorem stat in g t hat, under certain assumptions, every Laplacian process is t he limi t of some random

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... t ion d erived fro m a Poisson process. 1 The preparation of this paper was sponsorer1 (in part) by thc Officc of Naval Research. 'Loeve, Fonctions aleatoires d u second ordre, p. 320, in P . Levy, PrOccssus Stochastiques, Paris, 1948, Gauthiers·Villars edit. 32 (3) 3 We do not know if this rather obvious lemma is Or is not already known.