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This is the Exam of Statistical Science which includes Applied Statistics, Independent, Medals Winners, Following List, Medals, Proportion, Russia, China, Australia etc. Key important points are: Advanced Probability, Subsets, Independent Random Variables, Symmetric, Exists, Borel Cantelli Lemma, Martingale Probability, Arbitrage, Finite, Space
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Friday 30 May 2008 9.00 to 12.
Attempt FOUR questions. There are SIX questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Define the term σ-field. Let (Fr : r ∈ R) be a collection of σ-fields of subsets of Ω. Show that ∩r Fr is a σ-field, but that ∪r Fr need not be. Show that there exists a smallest σ-field of Ω containing every Fr.
Let X 1 , X 2 ,... be independent random variables on (Ω, F, P ). Define the tail σ- field of the Xi, and show that every event in the tail σ-field has probability either 0 or
Let Sn = X 1 + X 2 +... + Xn. Show that the events { lim inf n→∞ Sn/
n 6 −x
lim sup n→∞
Sn/
n > x
lie in the tail σ-field for every x ∈ R.
Suppose further that the Xi are symmetric (in that Xi and −Xi have the same distribution), and that there exists c ∈ R such that P (|Xi| 6 c) = 1 for all i. Show that
|Sn| 6
c infinitely often
2 (a) State and prove the two Borel–Cantelli lemmas.
(b) Show that the N (0, 1) distribution function Φ and density function φ satisfy
1 − Φ(x) ∼
φ(x) x
as x → ∞.
(c) Let X 1 , X 2 ,... be independent N (0, 1) random variables on (Ω, F, P ). Show that
P
lim sup n→∞
X n^2 log n
Advanced Probability
5 (a) Define the term ‘uniformly integrable’. Let Z be integrable and Fn a filtration. Show that the sequence Xn = E(Z|Fn), n > 1, is uniformly integrable.
(b) For x ∈ [0, 1), define for non-negative integers k, n,
bn(x) = k 2 −n^ if k 2 −n^6 x < (k + 1)2−n^.
Let f : [0, 1] → R be integrable, and let U be uniformly distributed on [0, 1]. Show that Xn = E(f (U )|Fn) defines a uniformly integrable martingale with respect to the filtration Fn = σ (bn(U )). Let fn be the step function on [0, 1] given by
fn(x) = 2n
∫ (^) bn(x)+2−n
bn(x)
f (u)du.
Show that fn(x) → f (x) for almost every x, and
∫ (^1)
0
|fn(u) − f (u)|du → 0 as n → ∞.
6 Define a standard Brownian motion B = (Bt : t > 0). Give a careful statement of the Strong Markov Property. Set
Mt = sup {Bs : 0 6 s 6 t}.
Prove that P (Mt > m, Bt 6 x) = P (Bt > 2 m − x)
for t > 0, m > 0, and x 6 m.
Deduce that Mt has the same law as |Bt|.
For x > 0, let Tx = inf{t : Bt > x}. Show that Tx has the same law as (x/B 1 )^2 , and calculate the density function of Tx.
Advanced Probability
