MATH 625 Homework V
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MATH 625 Homework IV
1. Proof. (a) Let x ∈ [0, 1] and δ > 0, we want to show that there exist n such that nλ mod 1 ∈ (x−δ, x+δ), for which it suffices to show that there exists n0 and y ∈ (−δ, δ) such that n0λ ≡ y (mod 1). Then take n to be an appropriate multiple of n0. To prove the reduced proposition, choose m such that 10−m < δ. Note that there exist n1 < n2 such that (n1λ mod 1) and (n2λ mod 1) have the same f...
متن کاملMATH 625 Homework VI
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. Su...
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