Stochastic Processes

a₁ absolutely continuous according to Theorem Baire function Borel field Borel set Brownian characteristic function closed linear manifold conditional expectation conditional probability continuous parameter converges with probability corresponding defined definition density discrete parameter discussion equation ergodic ergodic set example exists fact function F Gaussian given Hence hypothesis implies independent increments independent random variables inequality infinite integrand interval large numbers law of large Lebesgue measure lemma limit Markov process martingale matrix measurable sets measurable with respect measure-preserving metrically transitive mutually independent random non-negative orthogonal increments P₁ probability measure proof prove respect to F sample functions satisfied semi-martingale sequence set of probability spectral distribution function stationary processes stationary wide sense step functions stochastic integral stochastic process strictly stationary subset suppose t₁ Theorem 4.1 transition function true uniformly integrable values vanishes x₁