On the Characters and the Plancherel Formula of Nilpotent Groups ...

326 DUDLEY
It has been shown [A that for T an interval, hypothesis (c) of
Theorem 7.1 can be replaced by any of several conditions, of which
the best ([4J p. 186, 3”) seems to be
f 2k’2[,(1/29’/” < co.
k=l
But this condition is easily shown to imply Fernique’s.
Next we discuss random Fourier series and the work of Kahane [IO].
Let {xt , t E R) be a Gaussian process, stationary and periodic of
period 27r. [Note: Fernique’s counterexamples showing that Theo-
rem 7.1 (c) cannot be improved are all of this type, so the additional
hypotheses do not change that situation.) We assume xt is continuous
in probability and that Ex, z 0. It is then well known and not hard
to prove that a version of xt is given by
q(u) = 9 + f f3,(& sin nt + 7n cos nt),
n-1
(1”)
where the &(o) and Q( w ) are all independent, normalized Gaussian
random variables and the /3, are nonnegative constants, C &a < 00.
(Conversely, any such series (1”) defines a process of the given type.)
Kahane [JO] assumes /3,, = 0, which does not affect the sample
continuity.
Let
11.+,‘+1
ti2 = c &2.
s--e'+1
(Note: ti are not values of t!) Kahane ([IO], p. 2, Theoremes 3, 4)
proves the
THEOREM. The condition C& t, < co is necessary for sample
continuity or boundedness of xt and, if the ta are decreasing, also s@icient
(even for almost sure uniform convergence of ( 1”)).
Neither half of the above theorem will be proved here, and I
doubt that the methods of this paper would give such a complete
result. However, it will be shown that Conjecture 3.3 holds to the
extent that Kahane’s rather sharp result applies. Also we shall treat
some additional cases where Kahane’s theorem does not apply but
the conjecture still holds.
PROPOSITION 7.2. Suppose t, > t, 2 - -- and C t, < 00. Let S
be the set of all xI in H. Then r(S) < 2.