# Jurg Nievergelt, etc.'s Computer approaches to mathematical problems PDF

0 as m — 00, then ES= ES o . 6) Proof. First assume T < m for some integer m. Then ES, = E(Sol{L=ol) + E(Sll(T=i^) + ... + E(S m l {s=m) ) = E(Xo 1 (T, )) + E(X, 1 (,,)) + ...

Write Sm , n =Z l , n +— +Z,„ n for m>,1,So. 9) n Var X^ (n) = [ nt] Var Z„ n — [nt] ( (1 — 1\1 µ )Z) t, 7 n and a slight modification of the FCLT, with no significant difference in proof, implies that {X;n ) } and, therefore, {X;n°} converges in distribution to a Brownian motion with drift µ/Q and diffusion coefficient of I that starts at the origin. Let {X'} be a Brownian motion with drift It and diffusion coefficient a 2 starting at x. Then {W = (X, — x)/Q} is a Brownian motion with drift p/a and diffusion coefficient of!

From the FCLT, it follows that F sup IF(t) — F(t)I -- 0 in probability as n -• oo. 14) is also almost sure (Exercise 8). This stronger result is known as the Glivenko-Cantelli lemma. 13 STOPPING TIMES AND MARTINGALES An extremely useful concept in probability is that of a stopping time, sometimes also called a Markov time. }, defined on some probability space (0, F, P). Stopping times with respect to {X„} are defined as follows. Denote by . „ the sigmafield Q{X .... , X„} comprising all events that depend only on {X0 , X 1 .....