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

324 DUDLEY
since, e.g., Conjecture 5.9 is false if EV is replaced by EW, and Y and
EW are no functions of each other over a reasonable range.
We have not evaluated EV(Oc {b,)) if A({b,}) = + co, although
then for (b,} bounded we have EW(Oc) = - 1. Thus it is conceivable
that Conjecture 5.9 could hold even for EV < - 4, but it seems
unlikely.
7. PROCESSES ON EUCLIDEAN SPACES
In this section we apply Theorem 3.1 to Gaussian processes over
a finite-dimensional Euclidean parameter set, e.g., the usual one
dimensional “time”. Conjecture 3.3 is also verified in certain cases.
Since any compact Banach ball is a continuous image of the unit
interval,4 our hypotheses in general do not restrict the geometry of
the Banach balls in H which arise, and we do not try to evaluate their
volumes.
THEOREM 7.1 (Fernique [7], [7a] for T = cube). Suppose (x1, t E T}
is a Gaussian process where T is a bounded subset of Rk. Suppose q.~
is a nonnegative real-valued function such that
(a) Ej~~--x~1~ 0 such that
N(T, 6) < A/Sk for all s>o
(see [I2], Section 3, I, p. 20; cf. also Lemma 5.6 above). (b) and (c)
imply ~(8) J 0 as 6 J, 0.
For any E > 0 let
s = F(E) = sup {t : q(t) <