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A portion of an m.phil. Thesis in statistical science, focusing on advanced probability theory. It includes questions related to doob's maximal inequality, kolmogorov's inequality, laplace transforms, and the convergence of random variables. Students studying probability theory, stochastic processes, or related fields will find this document useful for understanding these concepts.
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Thursday 27 May, 2004 1.30 to 4.
Attempt FOUR questions. There are six questions in total.
The questions carry equal weight.
1 a) State and prove Doob’s maximal inequality.
b) Let ξ 1 , ξ 2 ,... be independent random variables with Eξj = 0 and E [ξj 2 ] < ∞. Denote Sn = ξ 1 +... + ξn. Prove Kolmogorov’s inequality: for any x > 0,
max 0 ≤m≤n
|Sm| ≥ x
E [Sn^2 ] x^2
c) Let Xn be a martingale with X 0 = 0 and E [Xn^2 ] < ∞. Show that for all x > 0
max 0 ≤m≤n
Xm ≥ x
E [Xn^2 ] E [Xn^2 ] + x^2
[Hint: Consider the process (Xn + c)^2 and optimize w.r.t. c. ]
2 a) Let X be a non-negative random variable with E X^2 < ∞. Show that its Laplace transform, φX (λ) ≡ E e−λX^ , λ ≥ 0, is differentiable for any λ > 0 and has the following decomposition:
φX (λ) ≡ E e−λX^ = 1 − E X λ +
λ^2 + o(λ^2 )
as λ → 0.
b) Let Bt be a Brownian motion starting from 0. For a > 0, denote by T the exit time
from the interval (−a, a), ie, T = inf
t : Bt ∈/ (−a, a)
. Show that
φT (λ) =
cosh
a
2 λ
and deduce that ET = a^2 , VarT = 2a^4 /3.
5 a) Assume n “stars” are located in the interval [−n, n] on the real line. Their locations are independent, each being uniformly distributed in the interval. Each star has mass m > 0, and the gravitational constant is unity. The force which will be exerted on a unit mass at the origin (the field strength) is then
Fn =
∑^ n
j=
m sign(Xj ) Xj 2
where Xj is the coordinate of the j’th star and sign(x) is the usual sign function. Show that the distributions of Fn converge weakly as n → ∞.
b) Suppose that the inverse-square attraction in a) were replaced by an inverse p’th power attraction. Show that for p satisfying 0 < 1 /p < 2, one gets convergence to the stable law of index 1/p, ie.,
E exp
itFn
→ exp
−cp|t|^1 /p
, as n → ∞,
with some constant cp > 0 to be found.
c) Suppose that the attraction is as in a) but the number of stars in [−n, n] is random, namely,
Fn =
j=
m sign(Xj ) Xj 2
where M is a Poisson random variable with parameter n, idependent of all Xj. Find the corresponding weak limit of the sequence Fn.
6 a) Let sequences Xn and Yn of random variables converge in probability to random variables X and Y respectively. Show that Xn + Yn converges in probability to X + Y.
b) We say that a sequence of random variables converges weakly if their distributions converge weakly. Show by counterexample that the analogue of the statement in a) for weak convergence may not be true.
c) Let a sequence Xn converge weakly to a random variable X and let a sequence Yn converge weakly to a constant random variable Y. Show that Xn + Yn converges weakly to X + Y.
