Probability Models, Worksheet 2
Due on Monday, October 29, at 15:00. You may submit up to 24 hours late with a penalty
of 5% on your mark.
The worksheet has 9 problems:
(1) You have to choose and submit 3 problems from part A.
(2) You have to choose and submit 2 problems from part B.
(3) Each problem can give you a maximum of 20 points.
(4) You may submit more problems if you wish, in order to get feedback on your
solutions. However, you need to clearly identify for which problems you want the
marks to count. If you don’t, I will choose myself, and my decision will be nal.
Solutions to all problems will be available three days after the deadline on Study Direct.
PART A: Models
Problem A-1: Suppose Zn =)Z1.
(a) Suppose you have a sequence of real numbers cn!c as n!1. Show that
Zn +cn =)Z1 +c:
(b) Suppose you have a sequence of random variables Yn ! c almost surely, as
n!1. Show that
Zn +Yn =)Z1 +c:
(c) Show that in general, if Zn =)Z1 and Yn =)Y1 then it is not necessarily
true that Zn +Yn =)Z1 +Y1.
Problem A-2: Suppose that fUngn 1 is a sequence of i.i.d. Unif[0;1] random variables. Let Mn
denote the running maximum of this sequence, i.e.
Mn = max1 i nfUig= maxfU1;U2;:::;Ung:
(a) Show that Mn converges in probability to the random variable X1 1.
(b) Show that the sequence Xn = n(1 Mn) converges weakly to an exponential
distribution.
(c) Does Mn converge to 1 in L1(P)?
(d) Does Mn converge to 1 P-a.s.?
Problem A-3: (Insurance Modeling) Let Xn be a sequence i.i.d. claims that come to an insurance
company in order. The insurance model suggests that the number of claims that
come in T time periods, follow a NT Poisson( T) distribution, and the total
money the company needs to pay are
SN =NTXi=1Xi:
(a) Prove that the sequence
ZT = SN E(SN)pVarSN
satis es a central limit theorem (i.e. it converges weakly to a normal ran-
dom variable) as T !1. You will need the mean and variance of SN from
Worksheet 1 OR rst you need to nd the characteristic function of SN.
(b) Assume the company has M customers, each one paying annual premium c in
order to be insured. Assume T = 52 (i.e. the time periods are weeks), = 2,
and each damage Xi Unif[1;5]. Approximate the smallest premium they can
charge, if they want the probability that they have a pro t of at least 2M to
be 95%.
Problem A-4: Consider two independent sequences fXngn2N;fYngn2N of continuous random
variables and let n and n be the characteristic functions of Xn and Yn, respec-
tively. Let ; 2R and X;Y;Z be random variables. Suppose that
Xn D!X; Yn D!Y; Xn + Yn D!Z:
(a) Assume further that the weak limits X and Y will be independent and that
the characteristic function of Z is continuous at the origin. Show that Z =
X + Y.
(b) Find an example of two sequences Xn;Yn so that in general Z6= X + Y in
distribution.
Problem A-5: (Random number generators) Let Xi be an i.i.d. sequence of discrete uniform. ran-
dom variables, with mass function given by
PfXi = jg= 110; j = 0;1;:::;9:
De ne a sequence of random numbers Un in [0;1] by
Un =nXi=1Xi10i:
Think of Un as the random number 0:X1X2X3:::Xn. Prove that Un converges
weakly to some U1 and identify the limiting distribution. It might be convenient
to rst compute PfUn = j10ng. What are the possible values of j in that case?
Problem A-6: (Rescaled Random Walks) The sequence (Xn)n2N of i.i.d. random variables with
PfXn = 1g= 13 and PfXn = 1g= 23
forms the scaled random walk
Wn = 3p8nnXi=1Xi pn+ 2p8
Show that there is a random variable W1 such that Wn =) W1 and nd its
distribution.
PART B: Probability
Problem B-1: Suppose that the Xi’s is a sequence of i.i.d. random variables with a c.d.f.
FX(x) that satis es
limx!1x (1 FX(x)) = > 0;
for some > 0. Let Mn be the running maximum, i.e.
Mn = maxfX1;X2;:::;Xng:
This exercise shows that there is a non-degenerate weak limit for the sequence
n 1= Mn.
(i) Show that limn!1n(1 FX(n1= x) = x .
(ii) Show the limit
limn!1Pfn 1= Mn xg= e x
for any x> 0.
(iii) Show that when x 0 the above limit is 0.
(iv) Suppose that the Xi’s have the Cauchy distribution, with density func-
tion
fX(x) = 1 (1 +x2); x2R:
Find the value of so that n 1= Mn has a non-degenerate limiting dis-
tribution, and give the c.d.f. of the limit.
Problem B-2: LetfXngbe a sequence of i.i.d. random variables for which there exists a t> 0
so that MX(t) = E(etjXnj) 0
PfXn >ag e taE(etXn)
(ii) Let Sn = Pni=1Xi and let "> 0. Show that
PfSn >n( +")g inft>0n
e nt( +")(E(etX))no
:(iii) Show that
limn!1 1n logPfSn >n( +")g sup
t>0nt( +") logMX(t)o:
(iv) Prove the strong law of large numbers for Sn=n, that can be equivalently
stated that for any "> 0
Pnlimn!1Snn =2( "; +")o= 0:4
Problem B-3: (i) The Continuous Mapping Theorem states that if g is a continuous
real-valued function and Xn =)X1 then
g(Xn) =)g(X1):
Prove it using the de nition of weak convergence.
(ii) Let X be any random variable, and an a sequence of numbers that goes
to in nity. Show that
limn!1Xa
n=)0:
It might be easier to actually show that it converges to 0 in probability
rather than weakly. In fact, something stronger is true that is necessary
for the next of the problem:
(iii) Prove that if Xn =)X and an !1, then a 1n Xn =) 0. You can use
this for the remaining parts of the problem.
(iv) Let a Xi be a sequence of i.i.d. Poisson( ). The central limit theorem
states that p
n( Xn ) =)Z N(0; ):
Also, the strong law of large numbers suggests that Xn is close to for
large n. By Taylor expanding logx around x0 = , nd the weak limit
of the sequence p
n(log Xn log ):