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2.6. LLNANDCLT
141
Proof TheproofofKolmogorov’sstronglawisnontrivial–see,forexample,theorem8.3.5of
[Dud02]
Ontheotherhand,wecanproveaweakerversionoftheLLNveryeasilyandstillgetmostofthe
intuition
Theversionweproveisasfollows: IfX
1
,...,X
n
isIIDwithEX
2
i
<¥,then,forany>0,we
have
Pfj
¯
X
n
mjeg!0 as n!¥
(2.20)
(Thisversionisweakerbecauseweclaimonlyconvergenceinprobabilityratherthanalmostsure
convergence,andassumeafinitesecondmoment)
Toseethatthisisso,fixe>0,andlets
2
bethevarianceofeachX
i
RecalltheChebyshevinequality,whichtellsusthat
Pfj
¯
X
n
mje
g
E[(
¯
X
n
m)
2
]
e2
(2.21)
Nowobservethat
E[(
¯
X
n
m)
2
]=E
8
<
:
"
1
n
n
å
i=1
(X
i
m)
#
2
9
=
;
=
1
n2
n
å
i=1
n
å
j=1
E(X
i
m)(X
j
m)
=
1
n
2
n
å
i=1
E(X
i
m)
2
=
s
2
n
Herethecrucialstepisatthethirdequality,whichfollowsfromindependence
Independencemeansthatifi6=j,thenthecovariancetermE(X
i
m)(X
j
m)dropsout
Asaresult,n
2
ntermsvanish,leadingustoafinalexpressionthatgoestozeroinn
Combiningourlastresultwith(2.21),wecometotheestimate
Pfj
¯
X
n
mjeg
s
2
ne2
(2.22)
Theclaimin(2.20)isnowclear
Ofcourse,ifthesequenceX
1
,...,X
n
iscorrelated,thenthecross-producttermsE(X
i
m)(X
j
m)
arenotnecessarilyzero
Whilethisdoesn’tmeanthatthesamelineofargumentisimpossible,itdoesmeanthatifwewant
asimilarresultthenthecovariancesshouldbe“almostzero”for“most”oftheseterms
Inalongsequence,thiswouldbetrueif,forexample,E(X
i
m)(X
j
m)approachedzerowhen
thedifferencebetweeniandjbecamelarge
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.6. LLNANDCLT
142
Inotherwords,theLLNcanstillworkifthesequenceX
1
,...,X
n
hasakindof“asymptoticin-
dependence”,inthesensethatcorrelationfallstozeroasvariablesbecomefurtherapartinthe
sequence
Thisideaisveryimportantintimeseriesanalysis,andwe’llcomeacrossitagainsoonenough
Illustration Let’snowillustratetheclassicalIIDlawoflargenumbersusingsimulation
Inparticular,weaimtogeneratesomesequencesofIIDrandomvariablesandplottheevolution
of
¯
X
n
asnincreases
Belowisafigurethatdoesjustthis(asusual,youcanclickonittoexpandit)
ItshowsIIDobservationsfromthreedifferentdistributionsandplots
¯
X
n
againstnineachcase
ThedotsrepresenttheunderlyingobservationsX
i
fori=1,...,100
Ineachofthethreecases,convergenceof
¯
X
n
tomoccursaspredicted
Thefigurewasproducedbyillustrates_lln.jl,whichisshownbelow(andcanbefoundinthe
lln_cltdirectoryoftheapplicationsrepository)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.6. LLNANDCLT
143
The three e distributions s are e chosen n at t random m from a a selection stored d in the e dictionary
distributions
#=
Visual illustration of f the law of large e numbers.
@author : Spencer Lyon <spencer.lyon@nyu.edu>
References
----------
Based off the original python n file illustrates_lln.py
=#
using PyPlot
using Distributions
100
srand(42# reproducible results
# == Arbitrary y collection n of distributions == #
distributions {"student's t with 10 degrees of f freedom" => TDist(10),
"beta(2, 2)" => Beta(2.02.0),
"lognormal LN(0, 1/2)" => LogNormal(0.5),
"gamma(5, 1/2)" => Gamma(5.02.0),
"poisson(4)" => Poisson(4),
"exponential with lambda = 1" => Exponential(1)}
# == Create a a figure e and some axes == #
num_plots 3
fig, axes plt.subplots(num_plots, 1, figsize=(1010))
bbox [0.1.021..102]
legend_args {:ncol => 2,
:bbox_to_anchor => bbox,
:loc => 3,
:mode => "expand"}
subplots_adjust(hspace=0.5)
for ax in axes
dist_names collect(keys(distributions))
# == Choose e a a randomly selected distribution == #
name dist_names[rand(1:length(dist_names))]
dist pop!(distributions, name)
# == Generate n draws from the distribution == #
data rand(dist, n)
# == Compute sample mean at each n == = #
sample_mean Array(Float64, n)
for i=1:n
sample_mean[i] mean(data[1:i])
end
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.6. LLNANDCLT
144
# == Plot == #
ax[:plot](1:n, data, "o", color="grey", alpha=0.5)
axlabel LaTeXString("\$\\bar{X}_n\$ for \$X_i \\sim\$ $name")
ax[:plot](1:n, sample_mean, "g-", lw=3, alpha=0.6, label=axlabel)
mean(dist)
ax[:plot](1:n, ones(n)*m, "k--", lw=1.5, label=L"$\mu$")
ax[:vlines](1:n, m, data, lw=0.2)
ax[:legend](;legend_args...)
end
InfiniteMean WhathappensiftheconditionEjXj < ¥inthestatement oftheLLNisnot
satisfied?
Thismightbethecaseiftheunderlyingdistributionisheavytailed—thebestknownexampleis
theCauchydistribution,whichhasdensity
f(x)=
1
p(1+x2)
(x2R)
Thenextfigureshows100independentdrawsfromthisdistribution
Noticehowextremeobservationsarefarmoreprevalentherethanthepreviousfigure
Let’snowhavealookatthebehaviorofthesamplemean
Herewe’veincreasednto1000,butthesequencestillshowsnosignofconverging
Willconvergencebecomevisibleifwetakenevenlarger?
Theanswerisno
Toseethis,recallthatthecharacteristicfunctionoftheCauchydistributionis
f(t)=Ee
itX
=
Z
e
itx
f(x)dx=e
jtj
(2.23)
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.6. LLNANDCLT
145
Usingindependence,thecharacteristicfunctionofthesamplemeanbecomes
Ee
it
¯
X
n
=Eexp
(
i
t
n
n
å
j=1
X
j
)
=E
n
Õ
j=1
exp
i
t
n
X
j
=
n
Õ
j=1
Eexp
i
t
n
X
j
=[f(t/n)]
n
Inviewof(2.23),thisisjuste
jtj
Thus,inthecaseoftheCauchydistribution,thesamplemeanitselfhastheverysameCauchy
distribution,regardlessofn
Inparticular,thesequence
¯
X
n
doesnotconvergetoapoint
CLT
Nextweturntothecentrallimittheorem,whichtellsusaboutthedistributionofthedeviation
betweensampleaveragesandpopulationmeans
StatementoftheTheorem
Thecentrallimittheoremisoneofthemostremarkableresultsinall
ofmathematics
IntheclassicalIIDsetting,ittellsusthefollowing:IfthesequenceX
1
,...,X
n
isIID,withcommon
meanmandcommonvariances
2(
0,¥),then
p
n(
¯
X
n
m)
d
!N(0,s
2
as n!¥
(2.24)
Here
d
!N(0,s2)indicatesconvergenceindistributiontoacentered(i.e,zeromean)normalwith
standarddeviations
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.6. LLNANDCLT
146
Intuition ThestrikingimplicationoftheCLTisthatforanydistributionwithfinitesecondmo-
ment,thesimpleoperationofaddingindependentcopiesalwaysleadstoaGaussiancurve
Arelativelysimpleproofofthecentrallimittheoremcanbeobtainedbyworkingwithcharacter-
isticfunctions(see,e.g.,theorem9.5.6of[Dud02])
Theproofiselegantbutalmostanticlimactic,anditprovidessurprisinglylittleintuition
InfactalloftheproofsoftheCLTthatweknowaresimilarinthisrespect
Whydoesaddingindependentcopiesproduceabell-shapeddistribution?
PartoftheanswercanbeobtainedbyinvestigatingadditionofindependentBernoullirandom
variables
Inparticular,letX
i
bebinary,withPfX
i
=0g=PfX
i
=1g=0.5,andletX
1
,...,X
n
beindepen-
dent
ThinkofX
i
=1asa“success”,sothatY
n
=
å
n
i=1
X
i
isthenumberofsuccessesinntrials
ThenextfigureplotstheprobabilitymassfunctionofY
n
forn=1,2,4,8
Whenn=1,thedistributionisflat—onesuccessornosuccesseshavethesameprobability
Whenn=2wecaneitherhave0,1or2successes
Noticethepeakinprobabilitymassatthemid-pointk=1
Thereasonisthattherearemorewaystoget1success(“failthensucceed”or“succeedthenfail”)
thantogetzeroortwosuccesses
Moreover,thetwotrialsareindependent,sotheoutcomes“failthensucceed”and“succeedthen
fail”arejustaslikelyastheoutcomes“failthenfail”and“succeedthensucceed”
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.6. LLNANDCLT
147
(Iftherewaspositivecorrelation,say,then“succeedthenfail”wouldbelesslikelythan“succeed
thensucceed”)
Here,alreadywehavetheessenceoftheCLT:additionunderindependenceleadsprobability
masstopileupinthemiddleandthinoutatthetails
Forn=4andn=8weagaingetapeakatthe“middle”value(halfwaybetweentheminimum
andthemaximumpossiblevalue)
Theintuitionisthesame—therearesimplymorewaystogetthesemiddleoutcomes
Ifwecontinue,thebell-shapedcurvebecomesevermorepronounced
Wearewitnessingthebinomialapproximationofthenormaldistribution
Simulation1 SincetheCLTseemsalmostmagical,runningsimulationsthatverifyitsimplica-
tionsisonegoodwaytobuildintuition
Tothisend,wenowperformthefollowingsimulation
1. ChooseanarbitrarydistributionFfortheunderlyingobservationsX
i
2. GenerateindependentdrawsofY
n
:=
p
n(
¯
X
n
m)
3. Usethesedrawstocomputesomemeasureoftheirdistribution—suchasahistogram
4. ComparethelattertoN(0,s
2)
Here’ssomecodethatdoesexactlythisfortheexponentialdistributionF(x)=e
lx
(PleaseexperimentwithotherchoicesofF,butrememberthat,toconformwiththeconditionsof
theCLT,thedistributionmusthavefinitesecondmoment)
#=
Visual illustration of f the central limit theorem
@author : Spencer Lyon <spencer.lyon@nyu.edu>
References
----------
Based off the original python n file illustrates_clt.py
=#
using PyPlot
using Distributions
# == Set parameters == #
srand(42# reproducible results
250
# Choice of n
100000 # Number of draws of Y_n
dist Exponential(1./2.# Exponential distribution, lambda = 1/2
mu, s mean(dist), std(dist)
# == Draw underlying RVs. Each row contains a draw of X_1,..,X_n == = #
data rand(dist, (k, n))
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.6. LLNANDCLT
148
# == Compute mean of each row, producing g k k draws of \bar X_n == #
sample_means mean(data, 2)
# == Generate e observations s of Y_n == #
sqrt(n) (sample_means .- mu)
# == Plot == #
fig, ax subplots()
xmin, xmax = -s, s
ax[:set_xlim](xmin, xmax)
ax[:hist](Y, bins=60, alpha=0.5, normed=true)
xgrid linspace(xmin, xmax, 200)
ax[:plot](xgrid, pdf(Normal(0.0, s), xgrid), "k-", lw=2,
label=LaTeXString("\$N(0, \\sigma^2=$(s^2))\$"))
ax[:legend]()
The
file
is
illustrates_clt.jl,
from
the
applications
repository
https://github.com/QuantEcon/QuantEcon.applications
Theprogramproducesfiguressuchastheonebelow
Thefittothenormaldensityisalreadytight,andcanbefurtherimprovedbyincreasingn
YoucanalsoexperimentwithotherspecificationsofF
Simulation2 Ournextsimulationissomewhat likethefirst, exceptthatweaimtotrackthe
distributionofY
n
:=
p
n(
¯
X
n
m)asnincreases
Inthesimulationwe’llbeworkingwithrandomvariableshavingm=0
Thus,whenn=1,wehaveY
1
=X
1
,sothefirstdistributionisjustthedistributionoftheunder-
lyingrandomvariable
Forn=2,thedistributionofY
2
isthatof(X
1
+X
2
)/
p
2,andsoon
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.6. LLNANDCLT
149
What weexpect isthat, regardlessofthedistribution oftheunderlyingrandomvariable, the
distributionofY
n
willsmoothoutintoabellshapedcurve
ThenextfigureshowsthisprocessforX
i
 f,wherefwasspecifiedastheconvexcombination
ofthreedifferentbetadensities
(Takingaconvexcombinationisaneasywaytoproduceanirregularshapeforf)
Inthefigure,theclosestdensityisthatofY
1
,whilethefurthestisthatofY
5
Asexpected,thedistributionsmoothsoutintoabellcurveasnincreases
Thefigureisgeneratedbyfilelln_clt/clt3d.jl,whichisavailablefromtheapplicationsrepos-
itoryhttps://github.com/QuantEcon/QuantEcon.applications
Weleaveyoutoinvestigateitscontentsifyouwishtoknowmore
IfyourunthefilefromtheordinaryJuliaorIJuliashell,thefigureshouldpopupinawindow
thatyoucanrotatewithyourmouse,givingdifferentviewsonthedensitysequence
TheMultivariateCase Thelawoflargenumbersandcentrallimittheoremworkjustasnicely
inmultidimensionalsettings
Tostatetheresults,let’srecallsomeelementaryfactsaboutrandomvectors
ArandomvectorXisjustasequenceofkrandomvariables(X
1
,...,X
k
)
EachrealizationofXisanelementofR
k
AcollectionofrandomvectorsX
1
,...,X
n
iscalledindependentif,givenanynvectorsx
1
,...,x
n
inR
k
,wehave
PfX
1
x
1
,...,X
n
x
n
g=PfX
1
x
1
gPfX
n
x
n
g
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
2.6. LLNANDCLT
150
(ThevectorinequalityXxmeansthatX
j
x
j
forj=1,...,k)
Letm
j
:=E[X
j
]forallj=1,...,k
TheexpectationE[X]ofXisdefinedtobethevectorofexpectations:
E[X]:=
0
B
B
B
@
E[X
1
]
E[X
2
]
.
.
.
E[X
k
]
1
C
C
C
A
=
0
B
B
B
@
m
1
m
2
.
.
.
m
k
1
C
C
C
A
=:m
Thevariance-covariancematrixofrandomvectorXisdefinedas
Var[X]:=E[(m)(m)
0
]
Expandingthisout,weget
Var[X]=
0
B
B
B
@
E[(X
1
m
1
)(X
1
m
1
)]  E[(X
1
m
1
)(X
k
m
k
)]
E[(X
2
m
2
)(X
1
m
1
)]  E[(X
2
m
2
)(X
k
m
k
)]
.
.
.
.
.
.
.
.
.
E[(X
k
m
k
)(X
1
m
1
)]  E[(X
k
m
k
)(X
k
m
k
)]
1
C
C
C
A
Thej,k-thtermisthescalarcovariancebetweenX
j
andX
k
WiththisnotationwecanproceedtothemultivariateLLNandCLT
LetX
1
,...,X
n
beasequenceofindependentandidenticallydistributedrandomvectors,eachone
takingvaluesinR
k
LetmbethevectorE[X
i
],andletSbethevariance-covariancematrixofX
i
Interpretingvectoradditionandscalarmultiplicationintheusualway(i.e.,pointwise),let
¯
X
n
:=
1
n
n
å
i=1
X
i
Inthissetting,theLLNtellsusthat
Pf
¯
X
n
!masn!¥g=1
(2.25)
Here
¯
X
n
!mmeansthatk
¯
X
n
!mk!0,wherekkisthestandardEuclideannorm
TheCLTtellsusthat,providedSisfinite,
p
n(
¯
X
n
m)
d
!N(0,Sas n!¥
(2.26)
Exercises
Exercise1 Oneveryusefulconsequenceofthecentrallimittheoremisasfollows
AssumetheconditionsoftheCLTasstatedabove
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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