MATH 625 Homework VI
نویسنده
چکیده
1. Proof. (a) It is known that all finite dimensional distributions of X(t) are multivariate normal with E(X(s)) = 0 and E(X(s)X(t)) = min{s, t}. It is easy to see that B(t) ∼ N(0, t), B(t′)−B(t) = X(s+ t′)−X(s+ t) ∼ N(0, t′− t) (t′ > t) and B(t0), B(t1) − B(t0), ..., B(tn) − B(tn−1) are independent. Hence all finite dimensional distributions of B(t) are multivariate normal with E(B(t)) = 0. Suppose that t < t′, then E(B(t)B(t′)) = E((X(t+ s)−X(s))(X(t′ + s)−X(s))) = E(X(t+ s)X(t′ + s))− E(X(s)X(t+ s))− E(X(s)X(t′ + s)) + E(X(s)) = (t+ s)− s− s+ s = t
منابع مشابه
MATH 625 Homework IV
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