Limit Theorems
Final Exam
Due 10:30 a.m., December 16
1. Prove that an infinitely divisible probability distribution µon Ris
supported on the rational numbers Q⊂R(i.e., µ(Q) = 1) if and only if
some translation of µby a rational number has the form eλ(M) for some
λ > 0 and some probability measure Mwhich is supported on Q.
2. Let {Xi}be a sequence of i.i.d. bounded random variables taking values
in the integer grid Z2. Let Sn=Pn
i=1 Xi. Prove that the sequence Snis a
recurrent Markov chain if and only if E[X1] = 0. Is this still true if we
allow the Xito be unbounded?
3. Can you give an example of a sequence of probability measures µnon R
whose characteristic functions φnconverge point-wise (as ntends to
infinity) to the function 1Z, where Zis the set of integers? What if we
replace 1Zwith 1A, where Ais the set of integers whose absolute values are
perfect squares?
4. Let µbe a probability measure on a measure space (Ω,F). Suppose A,
B, and Care σsubalgebras of Fand that Qais a family of measures on
(Ω,F) (indexed by a∈Ω) that gives a regular conditional probability for µ
given A. That is, for all S∈ F, the maps a→Qa(S) are Ameasurable;
Qx(A) = 1Afor all A∈ A; and for all F-measurable functions f, it is
µ-almost everywhere the case that Eµ[f|A] = Rf dQa. Suppose further
that the following hold:
(a) Ais µ-independent of B(i.e., for all A∈ A and B∈ B,
µ(A∩B) = µ(A)µ(B))
(b) Ais µ-independent of C
(c) For µ-almost all a, the algebras Band Care Qa-independent of each
other.
Prove that Ais µ-independent of the σalgebra generated by B ∪ C. Is this
still true without requirement (c)?
5. Let Ω be a (infinite dimensional with a countable basis) separable
Hilbert space and let Bbe the smallest field (not necessarily a σ-field)
containing all subsets of Ω of the form {f: (f, g)∈A}where g∈Ω and A
is a Borel subset of R. Let Fbe the σ-field generated by B.
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