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2013.09.27

Answer the following questions in order.

1. (25%=10+5+10)

(a) Give a probabilistic proof, using weak law of large numbers, of Weierstrass polynomial approximation theorem.

For (b) and (c), let (F_{n )} be a sequence of distribution functions such that the limit lim

J

^{xk dFn(x) = mk} exists for each k = 0,1,2, ...

n-too

(b) Show that (Fn ) is tight.

(c) Suppose that G is a distribution function concentrated on a compact set of lR and / xk dG(x) = mk, k = 0,1,2, ... Show that Fn :::::} G. (Hint. Apply approximation theorem.)

2. (25%=5+10+10) Let B_t

=

B(t), t ~ 0, be a one dimensional Brownian motion starting at O.

Define a new process Wt , t ~ 0, by Wo ^{= 0 and} Wt = tB(l/t) for t

>

O.

(a) Study the almost sure limit of Bn/n as the integer n -1 00.

(b) Derive the estimate

2 3 4 3

P ( sup IBt - Bnl

>

n ^{/ ) ::;} c n- ^/

tE[n,n+l]

for some constant c

>

O. Hint. Consider P ( _O<kS2sup _m IB (n

+ 2:) -

^B(n)1

^>

ⁿ 2 3^{/ )}

first.

(c) Use (b) to show that W_t is continuous at t

=

0 almost surely. Then verify that the process Wt is also a Brownian motion.

3. (25%=10+5+10) Let Zn be a martingale and T be a stopping time.

(a) Suppose that P(T ::; k) = 1 for some integer kEN. Show that E[Zo] = E[ZT] = E[Zk].

(b) Give an example of Zn such that E[Zo]

>

E[ZT] for some unbounded ^T.

(c) Show that E[Zo] = E[ZT] if P(T

<

00) = 1, E[jZTI]

<

00, and E[Zn1{T>n}] -1 0 as n -100.

4. (25%=9+6+10) Let Y_n be an irreducible Markov chain on a countable state space S. No proof is needed in (b).

(a) Give the definition that a state xES is transient, null recurrent and positive recurrent, respectively.

(b) When does the chain have a stationary distribution? Describe this distribution when exists.

(c) Discuss the transience or null/positive recurrence of the symmetric simple random walk on Zd, d = 1,2,3, ... Give an outline of the proof.