The ‘portmanteau theorem’ about convergence in distribution. Let .E; d/ be a metric space equipped with the -algebra E generated by the open sets, .Yn/n1 a sequence of random variables on a probability space . ;F; P/ taking values in .E; E /, and Q a probability measure on .E; E /. Prove the equivalence of the following four statements.

(a) Yn !d Q, i.e., limn!1 P.Yn 2 A/ D Q.A/ for all A 2 E with Q.@A/ D 0.

(b) lim supn!1 P.Yn 2 F/  Q.F / for all closed sets F  E.

(c) lim infn!1 P.Yn 2 G/  Q.G/ for all open sets G  E.

(d) limn!1 E.f ı Yn/ D EQ.f / for all bounded continuous functions f W E ! R.

Hints: (a))(b): If G" D ¹x 2 E W d.x;F/ < "º is the open "-neighbourhood of F , then Q.@G"/ D 0 for all except at most countably many ". (b), (c))(d): Suppose without loss of generality that 0  f  1, and approximate f from above by functions of the form 1

n Pn1 kD0 1¹f k=nº, and similarly from below.