STA 6467: Probability Theory II
Spring 2008
Problems for Assignment 2
1. Let Xn≥0 be independent for n≥1. The following are equivalent:
(i) P∞
n=1 Xn<∞a.s..
(ii) P∞
n=1P(Xn>1) + E[XnI[Xn≤1]]<∞.
(iii) P∞
n=1 EXn/(1 + Xn)<∞.
2. Suppose that P∞
n=1
σ2
n
n2=∞and without loss of generality that σ2
n≤n2for all n≥1.
Show that there exist independent random variables Xn,n≥1, with E[Xn]=0
and Var(Xn)≤σ2
n, for which Xn/n does not converge to 0 a.s. Conclude from this
that P∞
n=1 Xn/n does not converge a.s. and that n−1Pn
i=1 Xidoes not converge a.s.
(to any random variable, let alone to the constant 0).
If Xn,n≥1, are independent with E[Xn] = 0 and Var(Xn) = σ2
n, and if
P∞
n=1 σ2
n/n2<∞, then by Theorem 6.3.3 in the notes (or by a simple appli-
cation of Kolmogorov’s convergence criterion followed by Kronecker’s lemma),
P∞
n=1 Xn/n converges a.s. and n−1Pn
i=1 Xi→0 a.s. So the point of this
problem is to show that this result can easily fail when P∞
n=1 σ2
n/n2=∞.
3. Suppose that X1, X2, . . . are i.i.d. random variables with a symmetric distribution,
i.e., Xand −Xhave the same distribution. Then P∞
n=1(Xn/n) converges almost
surely iff E[|X1|]<∞.
4. Let {Xn, n ≥1}be nondegenerate i.i.d. L2random variables and let σ2= Var X1,
Xn=1
n
n
X
j=1
Xj, S2
n=1
n
n
X
j=1
(Xj−Xn)2, n ≥2.
Prove that Sn→σalmost surely and in L2, and that ESn→σ.
5. The L2weak law generalizes immediately to certain dependent sequences. Suppose
E[Xn] = 0 and E[XnXm]≤r(n−m) for m≤n, where r(k)→0 as k→ ∞. Show
that (X1+· · · +Xn)/n P
−→ 0 as n→ ∞.