Summary: Math­502. 1­th Homework. Due Friday, February 15, 2008.
1. Let {Xn}
n=1 be a sequence of r.v.'s defined in the same probability space. Show
that there exists a sequence of positive numbers {an} such that Xn
an
a.s.
0.
2. Find a sequence {Xn}
n=1 of r.v.'s such that E[Xn] 0, Var(Xn) 0 and {Xn}
n=1
does not converges a.s.
3. Let {Xn} be a sequence of i.i.d.r.v.'s with a N(0, 1) distribution. Show that
lim supn
Xn
2 log n
= 1 a.s. and lim infn
Xn
2 log n
= -1 a.s.
4. Let {an} be a sequence of positive numbers. Let {Xn} be a sequence of i.i.d.r.v.'s

Source: Arcones, Miguel A. - Department of Mathematical Sciences, State University of New York at Binghamton

Collections: Mathematics