Probability/ Limit Theorems
Final Examination
Due before Dec 19
Q1. For each n,{Xn,j};j= 1,2. . . , n are nmutually independent random variables taking
values 0 or 1 with probabilities 1 −pn,j and pn,j respectively. i.e
P[Xn,j = 1] = pn,j and P[xn,j = 0] = 1 −pn,j
If
lim
n→∞ sup
j
pn,j = 0,
then show that any limiting distribution of Sn=Xn,1+Xn,2+· · · +Xn,n is Poisson and
the limit exists if and only if
λ= lim
n→∞[pn,1+pn,2+· · · +pn,n ]
exists, in which case the limit is Poisson with parameter λ.
Q2. Is the exponential distribution with density
f(x) = e−xif x≥0 and 0 otherwise
infinitely divisible? If it is, what is its Levy-Khintchine representation? How about the
two sided exponential f(x) = 1
2e−|x|?
Q3. Let f(x) be an integrable function on [0,1] with respect to the Lebesgue measure.
For each nand j= 0,1, . . . , 2n−1 define for j2−n≤x≤(j+ 1)2−n
fn(x) = 2nZ(j+1)2−n
j2−n
f(x)dx
Show that limn→∞ fn(x) = f(x) a.e. with respect to the Lebsgue measure.
Q4. If X1, X2, . . . , Xn, . . . are independent random variables that are almost surely positive
(i.e. P[Xi>0] = 1) with E[Xi] = 1, show that
Zn=X1X2· · · Xn
is a martingale. What can you say about
lim
n→∞ Zn=Z?
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