A conditional dichotomy theorem for stochastic processes with independent increments

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M_t and N_t, respectively, and let the “time-jump” measures M and N be defined over [0, T] × $\text{R}$?{0} by M((t₁, t₂] × A) = M_t2(A) ? M_t1(A) and N((T₁, t₂] × A) = N_t2(A) ? N_t1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.     

Gaussian channels and the optimal coding

For the Gaussian channel Y(t) = Φ(ξ(s), Y(s); s ≦ t) + X(t), the mutual information I(ξ, Y) between the message ξ(·) and the output Y(·) is evaluated, where X(·) is a Gaussian noise. Furthermore, the optimal coding under average power constraints is constructed.