Probability 1
Exercises of week 11
Please solve the starred problems, the others will be discussed in the exercise classes.
Please hand in your solutions before 13/05/2014.
Exercise 1. We consider a sequence of real random variables (Xn )n≥1 and a real random variable
X. Prove the following statements.
(i) Xn → X almost surely if and only if for every > 0,
P lim sup{|Xn − X| ≥ } = 0.
n→∞
(ii) If for every > 0, the series
P
n
P [{|Xn − X| ≥ }] < ∞, then (Xn )n≥1 converges a.s. to X.
(iii) If the sequence of events ({|Xn − X| ≥ })n≥1 are independent, then Xn → X a.s. if and
only if for every > 0,
X
P [{|Xn − X| ≥ }] < ∞.
n
(iv) Xn → X almost surely if and only if the sequence (Yn )n≥1 , defined by Yn := supk≥n |Xk − X|,
converges in probability to zero.
Exercise 2. We consider a sequence of real random variables (Xn )n≥1 and a real random variable
X. Prove the following statements.
(i) If f : R → R is a continuous function, and Xn → X a.s., show that f (Xn ) → f (X) a.s. If f
is furthermore bounded, then E [f (Xn )] → E [f (X)].
(ii) Xn → X in probability if and only if for any sub-sequence of (Xn )n≥1 , we can extract a
further sub-sequence that converges almost surely to X.
(iii) If f : R → R is a continuous function, and Xn → X in probability, show that f (Xn ) → f (X)
in probability. If f is furthermore bounded, then E [f (Xn )] → E [f (X)].
Exercise 3. Prove the following statements are equivalent.
(i) Xn → X in probability.
(ii) There exists n ↓ 0, such that P [|Xn − X| > n ] ≤ n .
(iii) E [min(|Xn − X|, 1)] → 0.
Exercise 4. Let (Xn )n≥1 be i.i.d. with E [|X1 |] < ∞. Let Mn := max(X1 , · · · , Xn ). Prove that
n−1 Mn → 0 a.s.
Exercise 5. Consider a sequence of random variables (Xi )i∈N , let T be its tail σ-field. Remind
that
Pn by Kolmogorov’s zero-one law, T is trivial if (Xi )i∈N are independent. We denote that Sn =
i=1 Xi , and consider a deterministic sequence bn ↑ +∞. Are the following events in T ? Give
either a proof or a counter-example.
(i) {Xn → ∞}
(ii) {Sn converges as n → ∞}
(iii) {Xn > bn infinitely often}
(iv) {Sn > bn infinitely often}
*Exercise 6. We consider a sequence of independent random variables (Xi )i≥1 whose values are
in N.
(i) Show that (Xn )n≥1 converges in probability to zero, if and only if P [Xn > 0] tends to zero.
P
(ii) Show that (Xn )n≥1 converges a.s. to zero, if and only if n≥1 P [Xn > 0] < +∞.
(iii) Suppose that for every n ≥ 1, Xn has the Bernoulli distribution B(pn ), where pn ∈ (0, 1),
∀n ∈ N. For pn = n1 and pn = n12 respectively, determinate how (Xn )n≥1 converges to zero.
(In probability or in L1 or almost surely)
(iv) Suppose that for every n ≥ 1, Xn has the Poisson distribution P (αn ), where αn > 0, ∀n ∈ N.
For αn = n1 and αn = n12 respectively, determinate how (Xn )n≥1 converges to zero.
(v) Suppose that for every n ≥ 1, the law of Xn is given by P Xn = n2 = βn and P [Xn = 0] =
1 − βn . For βn = n1 , βn = n12 and βn = n13 respectively, determinate how (Xn )n≥1 converges
to zero.
*Exercise 7. Consider a sequence of positive random variables (Xi )i≥1 . For n ≥ 1, let Sn :=
Pi=n
i=1 Xi . Show that the sequence (Sn )n≥1 converges in probability if and only if it converges almost
surely.
Remark: the above statement is still true if we consider a sequence of independent random variables (Xi )i≥1 (but we do not suppose that they are positive).
*Exercise 8. With the help of the law of large numbers, calculate
(i)
Z
lim
n→+∞
f
[0,1]n
x1 + x2 + · · · + xn
n
dx1 · · · dxn .
where f : R → R is a bounded continuous function. We can consider a sequence of i.i.d.
random variables (Xi )i≥1 , each Xi has the uniform distribution on [0, 1].
(ii)
lim
n→+∞
X
e−αn
k≥0
(αn)k k
f ( ).
k!
n
where α ∈ R is strictly positive, and f : R → R is a bounded continuous function.
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