STAT4528: Probability and Martingale Theory

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STAT4528: Probability and Martingale Theory

1. (a) (i) De ne sets of real numbers as follows. Let An = (一 , 1] if n is odd, and

An = (一1, ] if n is even. Find lim supnAn and lim infnAn .

(ii) Show that if An t A of An l A then lim infnAn = lim supnAn = A.

(b) Show that if μ is a nite measure, there cannot be uncountably many disjoint sets

A such that μ(A) > 0.

(c) Let f be a Borel measurable function from R to R and a e R, and de ne g(x) = f(x + a). Show that

fdλ = hdλ

R R

in the sense that if one integral exists, so does the other, and the two are equal. (Start with indicators.)

2. (a) Let f and g be extended real-valued Borel measurable functions on ( , r), and

de ne

h(ω) =

where A is a set in r. Show that h is Borel measurable.

(b) Let X1, X2 , … be an i.i.d sequence such that E(lX1l) < o. Show that X1X2 + X2X3 + ... + XnXn+1 E(X1X2) as n goes to in nity. 3. Let Xn, n = 1, 2, . . ., and Y be random variables de ned on a common probability space ( , r, P). We assume that Xn, n ≥ 1 are independent and such that P (lXnl ≤ C) = 1, and EXn = 0, n = 1, 2, . . . where C > 0 is a constant.

Let gm = σ (Xn, n ≤ m) and go = σ (Xn, n ≥ 1).

(a) Show that

lim E (Y l gm) = E (Y l go ) , P 一 a.s.

(b) Assuming that Y is gm-measurable for a certain m ≥ 1, and such that ElY l < o, show that lim E (XnY) = 0 . (c) Using (2) or otherwise, show that n→o for arbitrary Y such that ElY l < o. 4. Let εn, n ≥ 1 be a sequence of independent and identically distributed random variables, such that Eεn = 0 and Eεn(2) < o. Let xn n ≥ be sequence of real numbers. Consider a linear regression problem Yn = axn + on, n ≥ 1, where a is an unknown slope parameter and Yn are observations. It is well known that that the Least Squares Estimator n based on the rst n observations Y1, . . . , Yn takes the form n (a) Identify a martingale (Mn) such that (M）n , where (M）n denotes the predictable quadratic variation of the martingale (Mn). Note, that you need to specify the ltration as well. (b) Show that n converges to a limit P-a.s. (c) Show that lim n = a, P 一 a.s. if and only if o xn(2) = o . n=1 (d) Let εn, n ≥ 1 be a sequence of independent and identically distributed random variables, such that Eεn = 0 and Elεnl < o. Is still the statement in (c) correct