hold:

The process y has sample paths in the RKHS H(R) with probability 1 if and only if the following two conditions hold:

H(K) H(R); (5)

n|| n||2R < , (6)

Proof. Suppose Equations (5) and (6) hold. Note first that the assumption H(K) H(R) implies the existence of the dominance operator L : H(R) H(R). According to Aronszajn [1], the operator L has the following representation: (Lf)(t) = (f, Kt)R, t [a,b].