Homework Set 4
1. Let Xnbe i.i.d. Bernoulli with parameter p6= 1/2. Consider the
random variable V=P∞
n=1 2−nXn. Show that there exists a set Dwith
Lebesgue measure 0 such that P(V∈D) = 1.
2. Let Xnbe i.i.d with P(X= 1) = p= 1 −P(X=−1), and Sn=
Pn
i=1 Xi.
Is {Sn= 0 infinitely often}a tail event? Supply arguments.
Show that P(Sn= 0 infinitely often) = 0 or 1.
Show that if p6= 1/2, P(Sn= 0 infinitely often) = 0. [Use Stirling’s
approximation n!∼√2πn nne−n]
3. Let X1, . . . , Xnbe i.i.d. observations from a parametric family
f(x;θ), and let Tnstand for the maximum likelihood estimator of θbased
on X1, . . . , Xn,n≥1. If Tn→Ta.s., show that Tmust be a constant.
4. If Anare events satisfying P∞
n=1 P(An∩Ac
n+1)<∞and P(An)→
0, then show that P(lim sup An) = 0. [Hint: With E= lim sup Anand
F= lim sup(Ac
n), show that E∩F⊂lim sup(An∩Ac
n+1) and use P(E)≤
P(Fc) + P(E∩F).]