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Probability and Measure 2017-2018 Example Sheet 3

Module

Probability and Measure

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Academic year: 2017/2018
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E. Breuillard Michaelmas 2017

Probability and Measure 3

You are not required to do the extra exercises marked with a star * at the end.

1 coin is tossed infinitely often, making an infinite sequenceω 1 ,... , ωn,.. .of heads or tails, i.ωi∈{H, T}. Show that every finite sequence of heads and tails (such asHHT T T HT) occurs infinitely often almost surely.

2.∑Let{Xn}n≥ 1 be a sequence of real random variables, such thatE(|Xn| 2 )<∞for eachnand n k=1E(X

2 k) =o(n

2 ) asn→+∞. Assume further thatE(Xn) = 0 for allnand that the variables

are pairwise uncorrelated, i. E(XiXj) = 0 ifi 6 = j. Show that 1 n

∑n k=1Xk converges to 0 in probability.

3μ,{μn}n≥ 1 be Borel probability measures onRwith distribution functionsF and{Fn}n≥ 1 respectively. Show thatμnconverges weakly toμif and only ifFn(x) converges toF(x) for every realx, whereFis continuous, and also if and only ifFn(x) converges toF(x) for Lebesgue almost everyx∈R.

4 a binomial random variableB(n, 12 ), e. Xnis the number of heads obtained after

tossing a fair coinntimes. Use the Stirling formula (n!enn−n−

1 2 →

√

2 π) to show that √ nP(Xn=k) = 2e−2(k−n/2)

2 /n /

√

2 π+o(1)

asn→+∞uniformly overkwhen (k−n/2)/

√

nremains bounded. Deduce that (Xn−E(Xn))/

√

n converges in distribution to a gaussianN(0, σ 2 ) withσ 2 = 14.

5 Scheff ́e’s lemma : let (fn:n∈N) be a sequence of integrable functions and suppose that fn→f a. for some integrable functionf. Show that, if‖fn‖ 1 → ‖f‖ 1 , then‖fn−f‖ 1 →0. Deduce that ifXn andX are real random variables whose law has a density with respect to Lebesgue measurefnandfrespectively and iffnconverges pointwise tof, thenXnconverges to Xin distribution.

1

2

6 a random variable and let 1≤p <∞. Show that, ifX∈Lp(P), thenP(|X| ≥λ) = O(λ−p) asl→∞. Prove the identity

E(|X|p) =

∫∞

0

pλp− 1 P(|X|≥λ)dλ

and deduce that, for allq > p, ifP(|X|≥λ) =O(λ−q) asl→∞, thenX∈Lp(P).

7μ,{μn}n≥ 1 be Borel probability measures onRand assume thatμnconverges weakly toμ. Show that one can find some probability space (Ω,F,P) and random variablesX,{Xn}n≥ 1 such thatXhas lawμ,Xnhas lawμnandXn→Xalmost surely asn→+∞. Can theXnbe chosen to be independent?

8 1 ,... , Xnbenreal random variables withE(|Xi| 2 )<∞fori= 1,... , n. The covariance matrix var(X) = (cij: 1≤i, j≤n) ofXis defined by

cij= cov(Xi, Xj) :=E[(Xi−E(Xi))(Xj−E(Xj)].

Show that var(X) is a non-negative definite matrix.

9 (Xn:n∈N) be an identically distributed sequence withE(|X 1 | 2 )<∞. Show thatnP(|X 1 |> ε

√

n)→0 asn→∞, for allε >0. Deduce thatn− 1 / 2 maxk≤n|Xk|→0 in probability.

10 (Xn:n∈N) be an identically distributed sequence of real random variables withE(|X 1 | 2 )< ∞. Show that E(max k≤n

|Xk|)/

√

n→0 as n→∞.

11 a uniformly integrable sequence of random variables (Xn:n∈N) such that bothXn→ 0 a. andE

(

supn|Xn|

)

=∞.

12{An}n≥ 1 be a sequence of events, which are pairwise weakly independent inthe sense that there is someC≥1 such thatP(Ai∩Aj)≤CP(Ai)P(Aj) for every two distincti 6 =j. Assume that

∑

n≥ 1 P(An) = +∞. Show thatP(lim supAn)>0. Hint: LetSn=

∑n k=1 1 Akand show thatYn=

Sn E(Sn)is bounded inL

2 hence uniformly integrable.

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Probability and Measure 2017-2018 Example Sheet 3

Module: Probability and Measure

8 Documents
Students shared 8 documents in this course

University: University of Cambridge

Was this document helpful?
E. Breuillard Michaelmas 2017
Probability and Measure 3
You are not required to do the extra exercises marked with a star * at the end.
1. A coin is tossed infinitely often, making an infinite sequence ω1, . . . , ωn, . . . of heads or tails,
i.e. ωi∈ {H, T }. Show that every finite sequence of heads and tails (such as HHT T T HT ) occurs
infinitely often almost surely.
2. Let {Xn}n≥1be a sequence of real random variables, such that E(|Xn|2)<∞for each nand
Pn
k=1 E(X2
k) = o(n2) as n→+∞. Assume further that E(Xn) = 0 for all nand that the variables
are pairwise uncorrelated, i.e. E(XiXj) = 0 if i6=j. Show that 1
nPn
k=1 Xkconverges to 0 in
probability.
3. Let µ,{µn}n≥1be Borel probability measures on Rwith distribution functions Fand {Fn}n≥1
respectively. Show that µnconverges weakly to µif and only if Fn(x) converges to F(x) for every
real x, where Fis continuous, and also if and only if Fn(x) converges to F(x) for Lebesgue almost
every x∈R.
4. Let Xnbe a binomial random variable B(n, 1
2), e.g. Xnis the number of heads obtained after
tossing a fair coin ntimes. Use the Stirling formula (n!enn−n−1
2→√2π) to show that
√nP(Xn=k) = 2e−2(k−n/2)2/n/√2π+o(1)
as n→+∞uniformly over kwhen (k−n/2)/√nremains bounded. Deduce that (Xn−E(Xn))/√n
converges in distribution to a gaussian N(0, σ2) with σ2=1
4.
5. Prove Scheff´e’s lemma : let (fn:n∈N) be a sequence of integrable functions and suppose that
fn→fa.e. for some integrable function f. Show that, if kfnk1→ kfk1, then kfn−fk1→0.
Deduce that if Xnand Xare real random variables whose law has a density with respect to
Lebesgue measure fnand frespectively and if fnconverges pointwise to f, then Xnconverges to
Xin distribution.
1

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