Extrema for Gap Power Series of Positive Real Part.

f*x. These pseudo-metrics determine topology 3, so f*x is continuous in topology 3. In essentially the same way, we prove that if AI > 0 and EM is the set of all x in (P having orthogonal increments, having E[x(t)2] < o for all t, and satisfying E [(x(t) x(s))2l < MIt sl for all real s and t, then for all f in L,(R) the mapping x. f*x is continuous in topology 3 on (PM. Other somewhat related topologies have been studied by Yu. V. Prokhorov,2 A. V. Skorokhod,3 and A. N. Kolmogorov.4 * Research supported in part by Army Research Office grant ARO-G-63. 1 Levy, Paul, Bull. d&s Sciences Math. (2), 84, 47-64 (1960). 2 Prokhorov, Yu. V., Theory of Probability and its Applications (S.I.A.M. translation), 1, 157214 (1956). 3 Skorokhod, A. V., Theory of Probability and its Applications (S.I.A.M. translation), 1, 261290 (1956); 2, 138-71(1957). 4 Kolmogorov, A. N., op. cit.,2 215-222.