Math/Stat 831 – Fall 2008
Homework 4
Due: November 13, 2008
1. Show that the difference of two identically distributed independent random variables
cannot have a uniform distribution on [-1,1].
2. Prove that if Xnconverges in distribution to a constant cthen Xn
P
−→ c.
3. Let X1, X2, . . . i.i.d with distribution function F(x). Denote the maximum of the first
nelement by Mn. Show that if
lim
x→∞
xα(1 −F(x)) = b
with fixed positive constants α, b then n−1/αMnconverges in distribution and identify
the limiting distribution.
Hint: you can use the convergence of the distribution functions to prove the weak limit.
4. (Ex. 2.3.13) Use characteristic functions to prove the following identity
sin(t)
t=∞
Y
k=1
cos µt
2k¶.
5. Prove the weak law of large numbers using characteristic functions.
Hint: use the result of the Problem 2.
6. (Ex. 2.4.9) Let X1, X2, . . . independent with the following distribution:
P(Xm=m) = P(Xm=−m) = 1
2m2,P(Xm= 1) = P(Xm=−1) = 1
2−1
2m2.
Let Sn=X1+· · · +Xn. Show that V ar(Sn)
n→2, but Sn
√n⇒N(0,1).
Bonus problem
1. Let X1, X2, . . . be i.i.d. Bernoulli(1/2) random variables. Let νndenote the index kwhen
we first have at least nzeros and nones among X1,. . . , Xk. Prove that νn−2n
√nconverges
in distribution and find the limit.