WEAK CONVERGENCE TO BROWNIAN MOTION

Let X_m=(X_m(t): t≥0) denote the continuous polygonalfunction whose vertices are where s_k =z_1+…+z_kand the Z_i's are independent random variables with Ez_i=0,vat z_i=σ_i~2;v_k=σ_1~2+…+σ_k~2 .Under standard condition onexpected values of ,itis shown that Xm converges weakly to Brownian motion in thetopology induced by the metric ρ(x,y) =wherex and y denote real continuous functions on [0,∞) and t log log tis taken to be 1 when t≤e~e.A comparison is made withSakhanenkos' similar conclusion.It is also shown thatconvergence,in probability,of the Skorokhod embedding times isa necessary condition for weak convergence to Brownian motion onthe unit interval under the supremum norm

WEAK CONVERGENCE TO BROWNIAN MOTION