(F.2) Moreover, if R
F.2. EQUIVALENT CONDITIONS FOR CONVERGENCE IN DISTRIBUTION 173 As the limit of the increasing functions F n ` (restricted to Q ), this function is also increasing:
G(q)≤G(q0) whenq, q0∈Qandq < q0.
We now claim thatG(q)↓0 as q↓ −∞and G(q)↑1 asq↑ −∞— this is the place where the assumption (iii) is used. Let ε >0. Then, since ϕ(0) = 1 andθ 7→ϕ(θ) is continuous atθ= 0 (by Proposition XII.6), there existsδ >0 such that
1−ϕ(θ)
< ε when |θ|< δ.
In particular we have
1 δ
Z δ
−δ
1−ϕ(θ)
dθ < ε.
By assumption (iii) we haveϕn(θ)→ϕ(θ) for allθ. Since we have
1−ϕn(θ)
≤1 +|ϕn(θ)| ≤2, the Bounded convergence theorem (Corollary VII.24) implies
1 δ
Z δ
−δ
1−ϕn(θ)
dθ −→
n→∞
1 δ
Z δ
−δ
1−ϕ(θ) dθ,
and we can conclude that there exists anNεsuch that forn≥Nεwe have 1
δ Z δ
−δ
1−ϕn(θ)
dθ < 2ε.
In view of Lemma F.3 this gives forn≥Nε
νn
R\[−R, R]
<2ε,
whereR= 2/δ. For the corresponding cumulative distribution functions this implies Fn(x)<2ε forx <−R
Fn(x)>1−2ε forx >−R.
The inequalities inherited by the subsequential limitG(q) = lim`→∞Fn`(q) are then G(q)≤2ε forq <−R
G(q)≥1−2ε forq >−R.
Sinceε >0 was arbitrary, this shows thatG(q)↓0 asq↓ −∞andG(q)↑1 asq↑ −∞.
The functionGis not yet the cumulative distribution function we are looking for: it is only defined on the rational numbers, and it is not necessarily right-continuous. But we can use it to define
Fe(x) = inf
q>x q∈Q
G(q)
(this in fact defines the smallest right continuous function aboveG). From the definition it is clear that ifq0, q00∈Qandq0< x < q00, then
G(q0)≤F(x)e ≤G(q00).
We leave it to the reader to check that thisFeis right continuous, increasing, and has limits 0 and 1 at −∞ and +∞, respectively. Therefore Fe is a cumulative distribution function of some probability measureνeonR.
The only remaining claim is that the subsequence (Fn`)`∈Nconverges toFepointwise in the set De ⊂Rof continuity points of Fe. Considerx∈ De and let ε > 0. Then by continuity ofFe atxwe have for someδ >0
Fe(x−δ)>Fe(x)−ε and Fe(x+δ)<F(x) +e ε.
Chooseq0∈Q∩(x−δ, x). Then by the definitionG(q0) = lim`→∞Fn`(q0) there exists some Lε such that for all`≥Lεwe have
Fn`(q0)> G(q0)−ε, which yields
Fn`(x)≥Fn`(q0)> G(q0)−ε≥Fe(x−δ)−ε >F(x)e −2ε.
174 F. CHARACTERISTIC FUNCTIONS Sinceε >0 was arbitrary, we get
lim inf
` Fn`(x)≥Fe(x).
By choosingq00∈Q∩(x, x+δ), one can similarly argue that lim sup
`
Fn`(x)≤Fe(x), and these two combined imply
`→∞lim Fn`(x) =F(x).e
This establishes the existence of subsequential limits (1), and finishes the proof.
Index
π-system,see also pi-system σ-algebra,see also sigma algebra p-integrable, 78
addition of functions,see also pointwise sum of functions
additivity countable, 8 finite, 13
almost sure,see alsoalmost surely event, 9
limit, 108 almost surely, 9 axiom of choice, 129 ball, 141
binary sequence, 135 binomial distribution, 13 Borel sigma algebra, 5 Borel-measurable function, 22 cardinality, 131
Cartesian product, 81, 128 Cauchy sequence
of square integrable random variables, 155 Cauchy-Schwarz inequality, 97
central limit theorem, 107 characteristic function, 120, 159 Chebyshev’s inequality, 111 closed set, 141
closed subspace, 156 complement, 127
complex valued random variable, 118 conditioned probability measure, 11 continuous distribution, 74, 95 continuous function, 141 convergence
almost surely, 108 inL1, 115, 154 in probability, 108
of a sequence of numbers, 138 convergence in distribution, 117, 125 convergence in law, 117, 125
convergent sequence, 141 countable set, 132 countably infinite, 132 counting measure, 9, 90
covariance, 98
cumulative distribution function, 18, 103 d-system, 143
De Morgan’s laws, 128 decreasing sequence
of numbers, 139 of sets, 129 density function, 74
joint, 95 marginal, 96 difference
of sets, 127 Dirac measure, 96 discrete metric, 141 disjoint, 127 disjoint union, 127 distribution
joint, 93
of a random variable, 23, 72 dominated convergence theorem, 68 Euclidean norm, 140
ev.,see also eventually event, ix, 1, 2
eventually, 44
expected value, ix, 55, 71 exponential distribution, 74 extended real line, 27, 137 Fatou’s lemma, 67
reverse, 67 finite measure, 9 finite measure space, 9 gaussian distribution, 74 geometric distribution, 13 Goddess of Chance, ix homeomorphism, 142 i.o.,see alsoinfinitely often image of a set under function, 128 improper Riemann integral, 70 increasing sequence
of numbers, 139 of sets, 129 independence, 41 175
176 INDEX ofσ-algebras, 39
of events, 40
of random variables, 40 indicator random variable, 24 inf,see alsoinfimum
infimum, 137, 138 infinitely often, 44 inner product
of square integrable random variables, 153 integrable
complex random variable, 118 integrable function, 63
integrable random variable, 63, 78 integral, 55
of a non-negative function, 59 of a simple function, 57 of an integrable function, 63 over subset, 69
intersection of sets, 127 Jensen’s inequality, 79 joint density, 95 joint law, 93
Kolmogorov’s strong law of large numbers, 116
law,see alsodistribution of a random variable, 23, 72 law of large numbers, 107 Lebesgue integral, 70 Lebesgue measure, 10, 90
d-dimensional, 10, 90 liminf
of events,see also eventually of sequence of numbers, 139 of sequence of sets, 130 limit, 141
of a decreasing sequence of sets, 129 of a sequence of numbers, 138 of an increasing sequence of sets, 129 limsup
of events,see also infinitely often of sequence of numbers, 139 of sequence of sets, 130 linearity
of integral, 56 lower limit
of sequence of numbers, 139 of sequence of sets, 130
marginal density,see also density function, marginal, 96
Markov process, 94 Markov’s inequality, 111
MCT,see also Monotone convergence theorem
measurable set, 7
space, 7, 8
measurable function, 21, 22 measurable space, 7 measure, 8
measure space, 9 moment, 78
monotone class, 38, 82, 143 Monotone class theorem, 82 monotone convergence
of integrals, 62 of measures, 13
Monotone convergence theorem, 61, 147 monotone sequence
of numbers, 139 of sets, 130 monotonicity
of integral, 56 of measures, 13
multiplication of functions,see also pointwise product of functions negative part
of a function, 63 norm
of square integrable random variable, 153 open set, 141
orthogonality
of square integrable random variables, 153, 156
outcome, ix
pi-system (π-system), 17
pointwise product of functions, 27 pointwise scalar multiple of functions, 27 pointwise sum of functions, 27
Poisson distribution, 12 positive part
of a function, 63 power set, 129
preimage of a set under function, 128 probability, ix
probability density, 95 probability mass function, 12 probability measure, 9 probability space, 9 product measure, 86
product of functions,see also pointwise product of functions
product sigma algebra, 83 random number, 22 random variable, ix, 21, 22 random walk, 54
Riemann integral, 70 sample space, ix
scalar multiplication of functions,see also pointwise scalar multiple of functions
INDEX 177 sequence
decreasing, 129, 139 increasing, 129, 139 monotone, 130, 139 of sets, 129
sigma algebra
generated by collection of events, 3 generated by random variables, 36 sigma algebra (σ-algebra), 2
sigma finite, 90 simple function, 30, 57
square integrable random variable, 97 staircase function, 31
standard machine, 56, 146 standard normal distribution, 74 strong law of large numbers, 110 subadditivity
countable, 14 finite, 13
sum of functions,see alsopointwise sum of functions
sup,see also supremum supremum, 137, 138 supremum norm, 141 sure event, 9
tail event, 48
tail sigma algebra, 48 tend
seeconverge, 138 total mass
of a measure, 9 truncated measure, 11 uncountable set, 135 uniform norm, 141
uniform probability measure continuous, 11
discrete, 10 on a finite set, 10 union of sets, 127 upper limit
of sequence of numbers, 139 of sequence of sets, 130 variance, 98
weak convergence, 117
weak law of large numbers, 110
Weierstrass’ approximation theorem, 112