Exercise 1∗. Let P(N) be the set of probabilities on (N,2N). We equip N with the digital metric d(i,j) := 1{i}(j) = 1{j}(i). Under such metric, any f : N → R is continuous (and |f| ≤ 12 implies 1-Lipschitz continuity). For notional convenience, we write μi := μ({i}) for μ ∈ P(N).
(a) (1 pt) (μn)n∈N ⊆ P(N) converges weakly to μ ∈ P(N) if and only if limn→∞ μin = μi for all i ∈ N.
(b) (1 pt) Let d1(μ, ν) := i∈N |μi − νi|. Show that P(N) under d1 is separable but not compact.
(c) (1pt)Letα:N→[0,1]satisfyα(i)≥α(i+1)fori∈Nandlimi→∞α(i)=0. LetSα bethe
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set of μ ∈ P(N) such that μ({i,i+1,…}) ≤ α(i) for i ∈ N. Show that Sα under d1 is compact.
In what follows, we let X a complete metric space with metric d.
Exercise 2. Prove the following statements:
(a) (0.5 pt) For any A ⊆ X and k ∈ N, fk(x) := exp(−kd(x, A)) is k-Lipschitz continuous.
(b) (0.5 pt) If f (x)μ(dx) = f (x)ν (dx) for all 1-Lipschitz continuous f with |f | ≤ 1, then XX
(c) (1 pt) If limn→∞ f (x)μn (dx) = f (x)μ(dx) for any f ∈ Cb (X) that is 1-Lipschitz and
|f| ≤ 1, then (μn)n∈N converges weakly to μ.
Exercise 3. (1 pt) Prove Theorem 14.4.
Exercise 4. Let λ be the Lebesgue measure on (R, B(R)). Define μn(B) := nλ([1 − n2 , 1 − n1 ] ∩ B)
for B ∈ B(R).
(a) (1 pt) Find the μ such that (μn)n∈N converges weakly to μ. Prove the convergence. (b) (1 pt) Compute
lim inf μn((0, 1)), lim sup μn([0, 1]),
lim inf μn((1, 2)), lim sup μn([1, 2]) n→∞ n→∞
μ((0, 1)), μ([0, 1]),
μ((1, 2)), μ([1, 2]).
Exercise 5. Let Y be a real-valued random variable and (Yn)n∈N be a sequence of real-valued
random variable.
(a) (0.5 pt) Show that (Yn)n∈N converges to Y in distribution if and only if limn→∞ E(f(Yn)) =
E(f(X)) for all f ∈ Cb(R).
(b) (0.5 pt) Show that if (Yn)n∈N converges to Y in probability, then (Yn)n∈N converges to Y in
distribution.
(c) (0.5 pt) Show that if (Yn)n∈N converges to a constant c ∈ R in distribution, then (Yn)n∈N
converges to c in probability.
(d) (0.5 pt) Give an example of (Yn)n∈N converging in distribution but not in probability.
Exercise 6. (1 pt) Prove Proposition 14.7.
Exercise 7. Let Y be a real-valued random variable, (Yn)n∈N and (Zn)n∈N be sequences of real- valued random variable. Suppose (Yn)n∈N converges to Y in distribution and (Zn)n∈N converges to a constant c ∈ R.
(a) (0.5 pt) Show that (ZnYn)n∈N converges to cY in distribution.
(b) (0.5 pt) Suppose c > 0. Show that (Yn/Zn)n∈N converges to Y/c in distribution.
(1 pts) Let (Yn)n∈N be a sequence of i.i.d. real-valued random variables with Y1 ∈ L2.
1n 1n 2
You can use what you have proved there.
Exercise 9. (2 pts) Let (Yn)n∈N be a sequence of i.i.d. real-valued random variables with E(Y1) = 1
and finite Var(Y1) = σ2 > 0. Let Sn := nk=1 Yk. Show that
where Z is N (0, 1) random variable. Hint: √1 (Sn − n) = √1 (√Sn + √n)(√Sn − √n).
Exercise 10. (2 pts) Let (Yn)n∈N be a sequence of i.i.d. real-valued random variables with E(Y1) = a > 0 and finite Var(Y1) = b2. Let (Zn)n∈N be another sequence of i.i.d. real-valued random variables that is also independent of (Yn)n∈N. Suppose P(Z1 = −1) = P(Z1 = 1) = 12. Find the limiting distribution of
√nnk=1 YkZk
nY asn→∞.
Exercise 11. (2 pts) Let U1 and U2 be independent U([0,1]) random variables. Let (Y,Z) be
R2-valued random variable with PDF f(Y,Z). For simplicity, we assume fY > 0. Define FY (r) := P(Y ∈ (−∞,r]), F−1(u) := inf{r ∈ R : F(r) ≥ u},
ShowthatP(R∈A,S∈B)=P(Y ∈A,Z∈B)foranyA,B∈B(R). (Comment: Wecan use monotone class theorem to extend the statement to P((R,S) ∈ D) = P((Y,Z) ∈ D) for any D ∈ B(R2). Generalization to random variables not necessarily having PDFs can be done using the notion of regular conditional distribution.)
Exercise 8.
sn := n − 1 Yk − n k=1
Show that Tn −−−→ Z, where Z is N(0,1) random variable. This is a continuation of HW2 Q8.
nk=1Yk−nE(Y1)
( Sn− n)−−−→Z,
FZ|Y (r|y) :=
(R,S):=F−1(U1),F−1 (U2|R)=F−1(U1),F−1 (U2|F−1(U1)).
fZ|Y (z|y) dz, FZ|Y (u|y) := inf{r ∈ R : FZ|Y (r) ≥ u}, Y Z|Y Y Z|Y Y
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