Probability And Measure
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Paper 1, Section II, H
comment(a) State and prove Fatou's lemma. [You may use the monotone convergence theorem without proof, provided it is clearly stated.]
(b) Show that the inequality in Fatou's lemma can be strict.
(c) Let and be non-negative random variables such that almost surely as . Must we have ?
Paper 2, Section II, H
commentLet be a measure space. A function is simple if it is of the form , where and .
Now let be a Borel-measurable map. Show that there exists a sequence of simple functions such that for all as .
Next suppose is also -integrable. Construct a sequence of simple -integrable functions such that as .
Finally, suppose is also bounded. Show that there exists a sequence of simple functions such that uniformly on as .
Paper 3, Section II,
commentShow that random variables defined on some probability space are independent if and only if
for all bounded measurable functions .
Now let be an infinite sequence of independent Gaussian random variables with zero means, , and finite variances, . Show that the series converges in if and only if .
[You may use without proof that for .]
Paper 4, Section II, 26H
commentLet be a probability space. Show that for any sequence satisfying one necessarily has
Let and be random variables defined on . Show that almost surely as implies that in probability as .
Show that in probability as if and only if for every subsequence there exists a further subsequence such that almost surely as .
Paper 1, Section II, 27K
comment(a) Let be a probability space. State the definition of the space . Show that it is a Hilbert space.
(b) Give an example of two real random variables that are not independent and yet have the same law.
(c) Let be random variables distributed uniformly on . Let be the Lebesgue measure on the interval , and let be the Borel -algebra. Consider the expression
where Var denotes the variance and .
Assume that are pairwise independent. Compute in terms of the variance .
(d) Now we no longer assume that are pairwise independent. Show that
where the supremum ranges over functions such that and .
[Hint: you may wish to compute for the family of functions where and denotes the indicator function of the subset
Paper 2, Section II,
commentLet be a set. Recall that a Boolean algebra of subsets of is a family of subsets containing the empty set, which is stable under finite union and under taking complements. As usual, let be the -algebra generated by .
(a) State the definitions of a -algebra, that of a measure on a measurable space, as well as the definition of a probability measure.
(b) State Carathéodory's extension theorem.
(c) Let be a probability measure space. Let be a Boolean algebra of subsets of . Let be the family of all with the property that for every , there is such that
where denotes the symmetric difference of and , i.e., .
(i) Show that is contained in . Show by example that this may fail if .
(ii) Now assume that , where is the -algebra of Lebesgue measurable subsets of and is the Lebesgue measure. Let be the family of all finite unions of sub-intervals. Is it true that is equal to in this case? Justify your answer.
Paper 3, Section II, 26K
commentLet be a probability measure preserving system.
(a) State what it means for to be ergodic.
(b) State Kolmogorov's 0-1 law for a sequence of independent random variables. What does it imply for the canonical model associated with an i.i.d. random process?
(c) Consider the special case when is the -algebra of Borel subsets, and is the map defined as
(i) Check that the Lebesgue measure on is indeed an invariant probability measure for .
(ii) Let and for . Show that forms a sequence of i.i.d. random variables on , and that the -algebra is all of . [Hint: check first that for any integer is a disjoint union of intervals of length .]
(iii) Is ergodic? Justify your answer.
Paper 4, Section II, K
comment(a) State and prove the strong law of large numbers for sequences of i.i.d. random variables with a finite moment of order 4 .
(b) Let be a sequence of independent random variables such that
Let be a sequence of real numbers such that
Set
(i) Show that converges in to a random variable as . Does it converge in ? Does it converge in law?
(ii) Show that .
(iii) Let be a sequence of i.i.d. standard Gaussian random variables, i.e. each is distributed as . Show that then converges in law as to a random variable and determine the law of the limit.
Paper 1, Section II, K
commentLet be an -valued random variable. Given we let
be its characteristic function, where is the usual inner product on .
(a) Suppose is a Gaussian vector with mean 0 and covariance matrix , where and is the identity matrix. What is the formula for the characteristic function in the case ? Derive from it a formula for in the case .
(b) We now no longer assume that is necessarily a Gaussian vector. Instead we assume that the 's are independent random variables and that the random vector has the same law as for every orthogonal matrix . Furthermore we assume that .
(i) Show that there exists a continuous function such that
[You may use the fact that for every two vectors such that there is an orthogonal matrix such that . ]
(ii) Show that for all
(iii) Deduce that takes values in , and furthermore that there exists such that , for all .
(iv) What must be the law of ?
[Standard properties of characteristic functions from the course may be used without proof if clearly stated.]
Paper 2, Section II, K
comment(a) Let for be two measurable spaces. Define the product -algebra on the Cartesian product . Given a probability measure on for each , define the product measure . Assuming the existence of a product measure, explain why it is unique. [You may use standard results from the course if clearly stated.]
(b) Let be a probability space on which the real random variables and are defined. Explain what is meant when one says that has law . On what measurable space is the measure defined? Explain what it means for and to be independent random variables.
(c) Now let , let be its Borel -algebra and let be Lebesgue measure. Give an example of a measure on the product such that for every Borel set , but such that is not Lebesgue measure on .
(d) Let be as in part (c) and let be intervals of length and respectively. Show that
(e) Let be as in part (c). Fix and let denote the projection from to . Construct a probability measure on , such that the image under each coincides with the -dimensional Lebesgue measure, while itself is not the -dimensional Lebesgue measure. Hint: Consider the following collection of independent random variables: uniformly distributed on , and such that for each
Paper 3, Section II, K
comment(a) Let and be real random variables such that for every compactly supported continuous function . Show that and have the same law.
(b) Given a real random variable , let be its characteristic function. Prove the identity
for real , where is is continuous and compactly supported, and where is a Lebesgue integrable function such that is also Lebesgue integrable, where
is its Fourier transform. Use the above identity to derive a formula for in terms of , and recover the fact that determines the law of uniquely.
(c) Let and be bounded random variables such that for every positive integer . Show that and have the same law.
(d) The Laplace transform of a non-negative random variable is defined by the formula
for . Let and be (possibly unbounded) non-negative random variables such that for all . Show that and have the same law.
(e) Let
where is a non-negative integer and is the indicator function of the interval .
Given non-negative integers , suppose that the random variables are independent with having density function . Find the density of the random variable .
Paper 4, Section II, K
comment(a) Let and be real random variables with finite second moment on a probability space . Assume that converges to almost surely. Show that the following assertions are equivalent:
(i) in as
(ii) as .
(b) Suppose now that is the Borel -algebra of and is Lebesgue measure. Given a Borel probability measure on we set
where is the distribution function of and .
(i) Show that is a random variable on with law .
(ii) Let and be Borel probability measures on with finite second moments. Show that
if and only if converges weakly to and converges to as
[You may use any theorem proven in lectures as long as it is clearly stated. Furthermore, you may use without proof the fact that converges weakly to as if and only if converges to almost surely.]
Paper 1, Section II, J
comment(a) Let be a real random variable with . Show that the variance of is equal to .
(b) Let be the indicator function of the interval on the real line. Compute the Fourier transform of .
(c) Show that
(d) Let be a real random variable and be its characteristic function.
(i) Assume that for some . Show that there exists such that almost surely:
(ii) Assume that for some real numbers , not equal to 0 and such that is irrational. Prove that is almost surely constant. [Hint: You may wish to consider an independent copy of .]
Paper 2, Section II, J
commentLet be a probability space. Let be a sequence of random variables with for all .
(a) Suppose is another random variable such that . Why is integrable for each ?
(b) Assume for every random variable on such that . Show that there is a subsequence , such that
(c) Assume that in probability. Show that . Show that in . Must it converge also in Justify your answer.
(d) Assume that the are independent. Give a necessary and sufficient condition on the sequence for the sequence
to converge in .
Paper 3, Section II, J
commentLet be the Lebesgue measure on the real line. Recall that if is a Borel subset, then
where the infimum is taken over all covers of by countably many intervals, and denotes the length of an interval .
(a) State the definition of a Borel subset of .
(b) State a definition of a Lebesgue measurable subset of .
(c) Explain why the following sets are Borel and compute their Lebesgue measure:
(d) State the definition of a Borel measurable function .
(e) Let be a Borel measurable function . Is it true that the subset of all where is continuous at is a Borel subset? Justify your answer.
(f) Let be a Borel subset with . Show that
contains the interval .
(g) Let be a Borel subset such that . Show that for every , there exists in such that
Deduce that contains an open interval around 0 .
Paper 4, Section II, J
commentLet be a measurable space. Let be a measurable map, and a probability measure on .
(a) State the definition of the following properties of the system :
(i) is T-invariant.
(ii) is ergodic with respect to .
(b) State the pointwise ergodic theorem.
(c) Give an example of a probability measure preserving system in which for -a.e. .
(d) Assume is finite and is the boolean algebra of all subsets of . Suppose that is a -invariant probability measure on such that for all . Show that is a bijection.
(e) Let , the set of positive integers, and be the -algebra of all subsets of . Suppose that is a -invariant ergodic probability measure on . Show that there is a finite subset with .
Paper 1, Section II, J
comment(a) Give the definition of the Borel -algebra on and a Borel function where is a measurable space.
(b) Suppose that is a sequence of Borel functions which converges pointwise to a function . Prove that is a Borel function.
(c) Let be the function which gives the th binary digit of a number in ) (where we do not allow for the possibility of an infinite sequence of 1 s). Prove that is a Borel function.
(d) Let be the function such that for is equal to the number of digits in the binary expansions of which disagree. Prove that is non-negative measurable.
(e) Compute the Lebesgue measure of , i.e. the set of pairs of numbers in whose binary expansions disagree in a finite number of digits.
Paper 2, Section II, J
comment(a) Give the definition of the Fourier transform of a function .
(b) Explain what it means for Fourier inversion to hold.
(c) Prove that Fourier inversion holds for . Show all of the steps in your computation. Deduce that Fourier inversion holds for Gaussian convolutions, i.e. any function of the form where and .
(d) Prove that any function for which Fourier inversion holds has a bounded, continuous version. In other words, there exists bounded and continuous such that for a.e. .
(e) Does Fourier inversion hold for ?
Paper 3, Section II, J
comment(a) Suppose that is a sequence of random variables on a probability space . Give the definition of what it means for to be uniformly integrable.
(b) State and prove Hölder's inequality.
(c) Explain what it means for a family of random variables to be bounded. Prove that an bounded sequence is uniformly integrable provided .
(d) Prove or disprove: every sequence which is bounded is uniformly integrable.
Paper 4, Section II, J
comment(a) Suppose that is a finite measure space and is a measurable map. Prove that defines a measure on .
(b) Suppose that is a -system which generates . Using Dynkin's lemma, prove that is measure-preserving if and only if for all .
(c) State Birkhoff's ergodic theorem and the maximal ergodic lemma.
(d) Consider the case where is Lebesgue measure on . Let be the following map. If is the binary expansion of (where we disallow infinite sequences of ), then where and are respectively the even and odd elements of .
(i) Prove that is measure-preserving. [You may assume that is measurable.]
(ii) Prove or disprove: is ergodic.
Paper 1, Section II, J
commentThroughout this question is a measure space and are measurable functions.
(a) Give the definitions of pointwise convergence, pointwise a.e. convergence, and convergence in measure.
(b) If pointwise a.e., does in measure? Give a proof or a counterexample.
(c) If in measure, does pointwise a.e.? Give a proof or a counterexample.
(d) Now suppose that and that is Lebesgue measure on . Suppose is a sequence of Borel measurable functions on which converges pointwise a.e. to .
(i) For each let . Show that for each .
(ii) Show that for every there exists a set with so that uniformly on .
(iii) Does (ii) hold with replaced by ? Give a proof or a counterexample.
Paper 2, Section II, J
comment(a) State Jensen's inequality. Give the definition of and the space for . If , is it true that ? Justify your answer. State and prove Hölder's inequality using Jensen's inequality.
(b) Suppose that is a finite measure space. Show that if and then . Give the definition of and show that as .
(c) Suppose that . Show that if belongs to both and , then for any . If , must we have ? Give a proof or a counterexample.
Paper 3, Section II, J
comment(a) Define the Borel -algebra and the Borel functions.
(b) Give an example with proof of a set in which is not Lebesgue measurable.
(c) The Cantor set is given by
(i) Explain why is Lebesgue measurable.
(ii) Compute the Lebesgue measure of .
(iii) Is every subset of Lebesgue measurable?
(iv) Let be the function given by
Explain why is a Borel function.
(v) Using the previous parts, prove the existence of a Lebesgue measurable set which is not Borel.
Paper 4, Section II, J
commentGive the definitions of the convolution and of the Fourier transform of , and show that . State what it means for Fourier inversion to hold for a function .
State the Plancherel identity and compute the norm of the Fourier transform of the function .
Suppose that are functions in such that in as . Show that uniformly.
Give the definition of weak convergence, and state and prove the Central Limit Theorem.
Paper 1, Section II, J
comment(a) Define the following concepts: a -system, a -system and a -algebra.
(b) State the Dominated Convergence Theorem.
(c) Does the set function
furnish an example of a Borel measure?
(d) Suppose is a measurable function. Let be continuous with . Show that the limit
exists and lies in the interval
Paper 2, Section II, J
comment(a) Let be a measure space, and let . What does it mean to say that belongs to ?
(b) State Hölder's inequality.
(c) Consider the measure space of the unit interval endowed with Lebesgue measure. Suppose and let .
(i) Show that for all ,
(ii) For , define
Show that for fixed, the function satisfies
where
(iii) Prove that is a continuous function. [Hint: You may find it helpful to split the integral defining into several parts.]
Paper 3 , Section II, J
comment(a) Let be a measure space. What does it mean to say that is a measure-preserving transformation? What does it mean to say that a set is invariant under ? Show that the class of invariant sets forms a -algebra.
(b) Take to be with Lebesgue measure on its Borel -algebra. Show that the baker's map defined by
is measure-preserving.
(c) Describe in detail the construction of the canonical model for sequences of independent random variables having a given distribution .
Define the Bernoulli shift map and prove it is a measure-preserving ergodic transformation.
[You may use without proof other results concerning sequences of independent random variables proved in the course, provided you state these clearly.]
Paper 4, Section II, J
comment(a) State Fatou's lemma.
(b) Let be a random variable on and let be a sequence of random variables on . What does it mean to say that weakly?
State and prove the Central Limit Theorem for i.i.d. real-valued random variables. [You may use auxiliary theorems proved in the course provided these are clearly stated.]
(c) Let be a real-valued random variable with characteristic function . Let be a sequence of real numbers with and . Prove that if we have
then
Paper 1, Section II,
commentWhat is meant by the Borel -algebra on the real line ?
Define the Lebesgue measure of a Borel subset of using the concept of outer measure.
Let be the Lebesgue measure on . Show that, for any Borel set which is contained in the interval , and for any , there exist and disjoint intervals contained in such that, for , we have
where denotes the symmetric difference .
Show that there does not exist a Borel set contained in such that, for all intervals contained in ,
Paper 2, Section II,
commentState and prove the monotone convergence theorem.
Let and be finite measure spaces. Define the product -algebra on .
Define the product measure on , and show carefully that is countably additive.
[You may use without proof any standard facts concerning measurability provided these are clearly stated.]
Paper 3, Section II, K
comment(i) Let be a measure space. What does it mean to say that a function is a measure-preserving transformation?
What does it mean to say that is ergodic?
State Birkhoff's almost everywhere ergodic theorem.
(ii) Consider the set equipped with its Borel -algebra and Lebesgue measure. Fix an irrational number and define by
where addition in each coordinate is understood to be modulo 1 . Show that is a measurepreserving transformation. Is ergodic? Justify your answer.
Let be an integrable function on and let be the invariant function associated with by Birkhoff's theorem. Write down a formula for in terms of . [You are not expected to justify this answer.]
Paper 4, Section II, K
commentLet be a sequence of independent identically distributed random variables. Set .
(i) State the strong law of large numbers in terms of the random variables .
(ii) Assume now that the are non-negative and that their expectation is infinite. Let . What does the strong law of large numbers say about the limiting behaviour of , where ?
Deduce that almost surely.
Show that
Show that infinitely often almost surely.
(iii) Now drop the assumption that the are non-negative but continue to assume that . Show that, almost surely,
Paper 1, Section II,
commentState Dynkin's -system -system lemma.
Let and be probability measures on a measurable space . Let be a -system on generating . Suppose that for all . Show that .
What does it mean to say that a sequence of random variables is independent?
Let be a sequence of independent random variables, all uniformly distributed on . Let be another random variable, independent of . Define random variables in by . What is the distribution of ? Justify your answer.
Show that the sequence of random variables is independent.
Paper 2, Section II,
commentLet be a sequence of non-negative measurable functions defined on a measure space . Show that is also a non-negative measurable function.
State the Monotone Convergence Theorem.
State and prove Fatou's Lemma.
Let be as above. Suppose that as for all . Show that
Deduce that, if is integrable and , then converges to in . [Still assume that and are as above.]
Paper 3, Section II,
commentLet be an integrable random variable with . Show that the characteristic function is differentiable with . [You may use without proof standard convergence results for integrals provided you state them clearly.]
Let be a sequence of independent random variables, all having the same distribution as . Set . Show that in distribution. Deduce that in probability. [You may not use the Strong Law of Large Numbers.]
Paper 4, Section II, K
commentState Birkhoff's almost-everywhere ergodic theorem.
Let be a sequence of independent random variables such that
Define for
What is the distribution of Show that the random variables and are not independent.
Set . Show that converges as almost surely and determine the limit. [You may use without proof any standard theorem provided you state it clearly.]
Paper 1, Section II, J
commentCarefully state and prove Jensen's inequality for a convex function , where is an interval. Assuming that is strictly convex, give necessary and sufficient conditions for the inequality to be strict.
Let be a Borel probability measure on , and suppose has a strictly positive probability density function with respect to Lebesgue measure. Let be the family of all strictly positive probability density functions on with respect to Lebesgue measure such that . Let be a random variable with distribution . Prove that the mapping
has a unique maximiser over , attained when almost everywhere.
Paper 2, Section II, J
commentThe Fourier transform of a Lebesgue integrable function is given by
where is Lebesgue measure on the real line. For , prove that
[You may use properties of derivatives of Fourier transforms without proof provided they are clearly stated, as well as the fact that is a probability density function.]
State and prove the almost everywhere Fourier inversion theorem for Lebesgue integrable functions on the real line. [You may use standard results from the course, such as the dominated convergence and Fubini's theorem. You may also use that where , converges to in as whenever
The probability density function of a Gamma distribution with scalar parameters is given by
Let . Is integrable?
Paper 3, Section II, J
commentCarefully state and prove the first and second Borel-Cantelli lemmas.
Now let be a sequence of events that are pairwise independent; that is, whenever . For , let . Show that .
Using Chebyshev's inequality or otherwise, deduce that if , then almost surely. Conclude that infinitely often
Paper 4, Section II, J
State and prove Fatou's lemma. [You may use the monotone convergence theorem.]
For a measure space, define to be the vector space of integrable functions on , where functions equal almost everywhere are identified. Prove that is complete for the norm