On packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits

0.18 MB·

by  Oleksandr Slutskyi

ModernStochastics:TheoryandApplications2(2015)371–389
DOI:10.15559/15-VMSTA44
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On packing dimension preservation by distribution

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functions of random variables with independent

] ˜
R Q-digits
P
.
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t OleksandrSlutskyi
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m
DepartmentofMathematicalAnalysisandDifferentialEquationsofDragomanov
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NationalPedagogicalUniversityofUkraine,Ukraine

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v [email protected](O.Slutskyi)
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3 Received:16October2015,Revised:13December2015,Accepted:13December2015,
1 Publishedonline:23December2015
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0 Abstract Thearticleisdevotedtofindingconditionsforthepackingdimensionpreservation
˜
. bydistributionfunctionsofrandomvariableswithindependentQ-digits.
1
The notion of “faithfulness of fine packing systems for packing dimension calculation”
0
is introduced, and connections between this notion and packing dimension preservation are
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1 found.
:
v Keywords Packingdimensionofaset,Hausdorff–Besicovitchdimensionofaset,
˜
i faithfulnessoffinepackingsystemforpackingdimensioncalculation,Q-expansionofreal
X
numbers,packing-dimension-preservingtransformations
r 2010MSC 28A78,28A80
a
1 Introduction
Let (M,ρ) be a metric space. Suppose that the Hausdorff–Besicovitch dimension
dim [8]iswelldefinedin(M,ρ).Atransformationf : M → M iscalleddimen-
H
sion-preservingtransformation[13]orDP-transformationif
dim f(E) =dim (E), ∀E ⊂M.
H H
Let G(M,dim ) be the(cid:0)set of(cid:1)all DP-transformationsdefined on (M,ρ). It is
H
easytoseethatGformsagroupw.r.t.thecompositionoftransformations.Itiswell
knownthatanybi-Lipschitztransformationbelongstothisgroup[8].However,Gis
©2015TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense.
www.i-journals.org/vmsta
372 O.Slutskyi
essentially wider than the group of all bi-Lipschitz transformations. In 2004, some
sufficientconditionsforbelongingofdistributionfunctionsofrandomvariablewith
independents-adicdigitstogroupGwasprovedbyG.Torbinetal.[2].Thereexist
alotofDP-functionsthatarenotbi-Lipschitz.
Sufficientconditionsfordistributionfunctionsofrandomvariableswithindepen-
dents-adicdigitstobeDP havebeenfoundbyG.Torbin[13]in2007.Thesecon-
ditionsweregeneralizedforQbyG.Torbin[14]andlaterforQ∗-andQ˜-expansions
byS.Albeverio,V.Koshmanenko,M.Pratsiovytyi,andG.Torbin[3,4].
Recently, G. Torbin and M. Ibragim proved rather general sufficient conditions
fordistributionfunctionsofrandomvariableswithindependentQ˜-digitstobeinDP-
class.Thenotionoffinecoveringsystemfaithfulnessfordim calculation[5]plays
H
animportantroleintheproofoftheseconditions.Thisnotiongivesusthepossibility
toconsidercoveringsbysetsfromsomefamilyΦandtobesurethata“dimension”
calculated in such a way is equal to dim . Faithfulness of the family of all s-adic
H
cylinders (if s is fixed) have been proven by Billingsley [6] in 1961. Faithfulness
of the family of Q-cylinders have been proven by M. Pratsiovytyi and A. Turbin
[16] in 1992, and faithfulness of the family of Q∗-cylinders (under the condition
of separation from zero of the corresponding coefficients) have been proven by S.
AlbeverioandG.Torbin[1]in2005.Itisnecessarytoremarkthatthelastresultcan
beeasilygeneralizedtoQ˜-expansionunderasimilarcondition.
In1982,C.Tricot[15]introducedthenotionofpackingdimensiondim .Thisdi-
P
mensionisinsomesensedualtotheHausdorff–Besicovitchdimension:thedefinition
ofdim ofasetF isbasedonε-coveringsofthisfigure,butthedefinitionofdim
H P
isbasedonε-packings(thecountablesetsofdisjointopenballsB (r ,c ),k ∈ N,
k k k
withradiir 6 εandcentersc ∈ F).Thepackingdimensionhasall“good”prop-
k k
erties of a fractal dimension, such as the countable stability. Therefore, proving or
disprovingsimilarresultsfordim isimportant.Forexample,weconsiderthegroup
P
ofpacking-dimension-preservingtransformations(orPDP-transformations).
Definition1.1. Thetransformationf issaidtobeaPDP-transformationif
∀E ⊂M, dim f(E) =dim (E).
P P
(cid:0) (cid:1)
Therearealotofproblemswithprovingofmanyconjecturesfordim because
P
workwithpackingsisessentiallymorecomplicatedthanworkwithcoverings[10].
These problemsare solving bit by bit. For example, M. Das [7] has proven the
Billingsleytheoremforpackingdimension;J.Li[9]obtainedsomesufficientcondi-
tionsfordistributionfunctionsofrandomvariableswithindependentQ˜-digitstobe
inPDP-class.Namely,J.Lihasproventhefollowingtheorem.
Theorem1.1. LetF bethedistributionfunctionofarandomvariableξ withinde-
ξ
pendentQ˜-representation.If inf q = q > 0 and inf p = p > 0, then F
i,j ij ∗ i,j ij ∗ ξ
preservesthepackingdimensionifandonlyif
h +h +···+h
1 2 k
limsup =1,
b +b +···+b
k→∞ 1 2 k
whereh =− nj p lnp andb =− nj p lnq .
j i=1 ij ij j i=1 ij ij
P P
PDP-transformations 373
In Remark 4.2 at the and of article [9], we read: “The conditions inf q =
i,j ij
q > 0 andinf p = p > 0playanimportantrolein theproofofthetheorem.
∗ i,j ij ∗
Openquestion:Whatcanwesayaboutthetopicifweremovetheseconditions?”
S.Albeverio,M.Pratsiovytyi,andG.Torbin[3]removedtheconditioninf p =
i,j ij
p >0inasimilarsituationforDP-transformations.
∗
In case of packingdimension,the approachof [3] is complicatedbecause it re-
quiresappropriateresultsaboutthefinepackingsystem faithfulnessforpackingdi-
mensioncalculation.Eventhedefinitionofthefinepackingsystemfaithfulnessisa
problembecausecentersofallballsinpackingsshouldbeinthesetthedimensionof
whichiscalculated.
Theaimofthispaperistoproposesomealternativedefinitionofthepackingdi-
mension,uncenteredpackingdimensionordimP(unc).Intheproposeddefinition,the
condition“thecentersofballsshouldbeinthefigurethedimensionofwhichiscal-
culated”inthedefinitionofdim isreplacedby“everyballshouldhaveanonempty
P
intersectionwiththefigure.”Weprovethat,insomewideclassofmetricspaces(in-
cluding Rn), the value of packing dimension with uncentered balls is matching to
the value of classical packing dimension. Introduction of the fine packing system
faithfulnessnotionis verysimple in the case of proposeddefinition.It allows usto
provefaithfulness(undertheconditionofseparationfromzeroofthecoefficients)of
a Q˜-cylindersystem and sufficientconditionsforthe distributionfunctionof a ran-
domvariablewithindependentQ˜-digitstobeinthePDP-class.Thecorresponding
theoremisthemainresultofthepaper.
Theorem1.2. Letinf q :=q .Supposethatq >0.Let
i,j ij min min
q
T := k :k ∈N,p < min ;
k
2
(cid:26) (cid:27)
T :=T ∩{1,2,...,k};
k
ln 1
B :=limsup j∈Tk pj .
k
k→∞ P
LetF be the distributionfunctionofa randomvariableξ with independentQ˜-
ξ
representation.ThenF preservesthepackingdimensionifandonlyif
ξ
dim µ =1;
P ξ
(B =0.
2 Packingdimension
Letusrecallthe definitionofpackingdimensionin the formgiven,forexample,in
[8].
Definition2.1. LetE ⊂ M andε > 0. A finite orcountablefamily{E }of open
j
ballsiscalledanε-packingofasetEif
374 O.Slutskyi
1. |E |6εforalli;
i
2. c ∈E, i∈N,wherec isthecenteroftheballE ;
i i i
3. E ∩E =∅foralli,j,i6=j.
i j
Remark2.1. Theemptysetofballsisapackingofanyset.
Definition2.2. LetE ⊂M,α>0,ε>0.Thentheα-dimensionalpackingpremea-
sureofaboundedsetE isdefinedby
Pα(E):=sup |E |α ,
ε i
(cid:26) i (cid:27)
X
where the supremum is taken over all at most countable ε-packings{E } of E (if
j
E =∅forallj,thenPα(E)=0).
j ε
Definition2.3. Theα-dimensionalpackingquasi-measureofasetE isdefinedby
Pα(E):= limPα(E).
0 ε
ε→0
Definition2.4. Theα-dimensionalpackingmeasureisdefinedby
Pα(E):=inf Pα(E ):E ⊂ E ,
0 j j
(cid:26) j (cid:27)
X [
wheretheinfimumistakenoverallatmostcountablecoverings{E }ofE,E ⊂M.
j j
Definition2.5. Thenonnegativenumber
dim (E):=inf α:Pα(E)=0
P
iscalledtheuncenteredpackingdimensio(cid:8)nofasetE ⊂M(cid:9).
3 Uncenteredpackingdimension
Definition3.1. LetE ⊂ M andε > 0. A finite orcountablefamily{E }of open
j
ballsiscalledanuncenteredε-packingofasetE if
1. |E |6εforalli;
i
2. E ∩E 6=∅;
i
3. E ∩E =∅foralli,j,i6=j.
i j
Remark3.1. Theemptysetofballsisanuncenteredpackingofanyset.
Definition 3.2. Let E ⊂ M, α > 0, ε > 0. Then the uncentered α-dimensional
packingpremeasureofaboundedsetE isdefinedby
Pα (E):=sup |E |α ,
ε(unc) i
(cid:26) i (cid:27)
X
wherethesupremumistakenoverallatmostcountableuncenteredε-packings{E }
i
ofE.
PDP-transformations 375
Definition3.3. Theuncenteredα-dimensionalpackingquasi-measureofa setE is
definedby
Pα (E):= limPα (E).
0(unc) ε(unc)
ε→0
Definition3.4. Uncenteredα-dimensionalpackingmeasureisdefinedby
Pα (E):=inf Pα (E ):E ⊂ E ,
(unc) 0(unc) j j
(cid:26) j (cid:27)
X [
wheretheinfimumistakenoverallatmostcountablecoverings{E }ofE,E ⊂M.
j j
Remark3.2. If(M,ρ) = R1 andα = 1,thentheα-dimensionalpackingmeasure
anduncenteredα-dimensionalpackingmeasurearetheLebesguemeasure.
Definition3.5. Thenonnegativenumber
dimP(unc)(E):=inf α:P(αunc)(E)=0 .
iscalledtheuncenteredpackingdimension(cid:8)ofasetE ⊂M. (cid:9)
Theorem3.1. Let(M,ρ)beametricspace.LetC ∈N.Ifforallr >0andforany
openballI with|I|=8r,thereexistatmostN(I)ballsI , i∈{1,..., N(I)}such
i
thatI ⊂I, i∈{1,...,N(I),|I |=r, i∈{1,...,N(I)},and N(I)≤C.Then
i i
dimP(unc)(E)=dimP(E).
Proof. Step1.LetusprovetheinequalitydimP(unc)(E)>dimP(E).
Bythedefinitionsandsupremumpropertywehave
Pα (E)>Pα(E).
r(unc) r
Bythelimitpropertyofinequalitieswehave
Pα (E)>Pα(E).
0(unc) 0
Hence,
Pα (E)>Pα(E).
(unc)
LetdimP(unc)(E)=α0.BythedefinitionofdimP(unc)(E)wehave
∀ε>0, Pα0+ε(E)=0.
(unc)
Therefore,
∀ε>0, Pα0+ε(E)=0,
0
and,consequently,
dim (E)6α .
P 0
Hence,itfollowsthatdimP(unc)(E)>dimP(E),whichisourclaim.
Step2.LetusshowthatdimP(unc)(E)6dimP(E).
IfdimP(unc)(E)=0,thenthestatementistrue.
376 O.Slutskyi
LetusconsiderthecasedimP(unc)(E)6=0.Fix0<t<s<dimP(unc)(E).
Sinces<dimP(unc)(E),wehave
Ps (E)=+∞,
(unc)
Ps (E)=+∞.
0(unc)
Therefore,
∀r >0, Ps (E)=+∞.
r(unc)
From this and from the supremum property, it follows that there is an uncentered
packingV :={E }ofthesetE with
i
|E |s >1. (1)
i
i
X
LetusdividethepackingV intoclasses
V := E :2−k−1 6|E |<2−k .
k i i
Letn bethenumberofballsV(cid:8).Wewillshowthat (cid:9)
k k
∃k0 :nk0 >2k0t 1−2t−s .
Toobtainacontradiction,supposethat (cid:0) (cid:1)
n <2kt 1−2t−s forallk.
k
Then (cid:0) (cid:1)
|E |s < 2−ks·n < 2−ks·2kt 1−2t−s = 1−2t−s · 2t−s k =1,
i k
i k k k
X X X (cid:0) (cid:1) (cid:0) (cid:1) X(cid:0) (cid:1)
whichcontradictsourassumption(1).
Therefore,such k exists. Letus considerV . We denotebyA ,A ,...,A
0 k0 1 2 nk0
theballsinV ,thatis,
k0
V ={A ,A ,...,A }.
k0 1 2 nk0
Fixr :=2−k0−1.ThentheradiusofanyAiislessthanr.LetTibeapointofAi
suchthatT ∈A ∩E.LetV′bethesetofballswiththecentersT andradiusr,that
i i i
is,
V′ = A′ :A′ =B(T ,r) .
i i i
Fix (cid:8) (cid:9)
V∗ = A∗ :A∗ =B(T ,4r) .
i i i
LetusdividethesetV′intoc(cid:8)lassesK1,K2,...,Kl(cid:9)asfollows.
1. LetustakeaballA′ =A′ andputitinK togetherwithallotherballsA′ ∈
V′suchthatA′ ∩Aj′1 6=∅1. 1 i
i j1
2. LetustakeanarbitraryballA′ ∈ V′\K andputitinK togetherwithall
otherballsA′ ∈V′\K suchjt2hatA′ ∩A′1 6=∅. 2
i 1 i j2
PDP-transformations 377
3. LetuscontinuethiswayuntilV′\(K ∪K ∪···∪K )6=∅.Sincethenumber
1 2 l
ofelementsinasetV′isafinite,wecanfindsuchanumberl.
Now suppose that the balls A′ and A′ intersect each other. In other words,
i j
ρ(T ,T )62r.Therefore,A ⊂A∗.
i j j i
TheradiusofA isgreaterthanr/2.Bythetheoremcondition,therearenomore
j
thanC disjointballswithradiusr/2inaballwithradius4r
Therefore,therearenomorethanC ballsinanyclassK .
i
Moreover,inthecasei < m,theballsA′ andA′ donotintersecteachother.
ji jm
Indeed,supposeotherwise.ThenA′ isinaclassK orinaclasswithnumberless
jm i
thani.
Hence,
V′′ = A′ ,A′ ,...,A′
j1 j2 jl
is a centered packing of a set E, a(cid:8)nd the t-volume of(cid:9)this packing is less than the
t-volumeoftheuncenteredpackingV nomorethanC times.Therefore,
k0
A′ji t >nk0 · 2−Ck0t >2k0t 1−2t−s · 2−Ck0t = 1−C2t−s.
V′′
X(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
Fromthisitfollowsthat
1−2t−s
Pt (E)> .
2−k0 C
Bytheinequality2−k0 <rweget
1−2t−s
Pt(E)> forallr>0.
r C
Consequently,asr →0,wegettheinequality
1−2t−s
Pt(E)> .
0 C
LetusshowthatPt(E)> 1−2t−s.Recallthedefinition
C
Pt(E)=inf Pt(E ):E ⊂ E ,
0 j j
(cid:26) j (cid:27)
X [
wheretheinfimumistakenoverallatmostcountablecoveringsE ofasetE.
j
Let {Ej} be an at most countable covering of E. Since dimP(unc)(E) > s,
thereisj0 suchthatdimP(unc)(Ej0) > s (bythecountablestabilityofthepacking
dimensiondimP(unc)).Inotherwords,wehave
Ps (E )=+∞,
(unc) j0
Ps (E )=+∞.
0(unc) j0
WeconcludebythepartofthetheoremalreadyprovedforE that
1−2t−s
Pt(E )>
0 j0 C
378 O.Slutskyi
and
1−2t−s
Pt(E )> .
0 j C
j
X
Butthepreviousinequalityistrueforanarbitrarycovering{E }ofasetE andfor
j
theinfimumforallcoverings.Therefore,
1−2t−s
Pt(E)>
C
and
dim (E)>t.
P
Sincet-dimP(unc)(E)canbeapproximatedby0,wegetdimP(E)>dimP(unc)(E),
whichcompletestheproof.
Corollary3.1. IfM =Rn,thendimP(unc)(E)=dimP(E).
Proof. Let B be a ball with radius 8r, B be a ball with radius r, and λ be the
8r r
n-dimensionalLebesguemeasure.Then
λ(B )=8n·λ(B ).
8r r
Therefore,we canputnomorethanC = 8n disjointballswithradiir in aball
withradius8r,whichcompletestheproof.
3.1 Packingdimensionwithrespecttothefamilyofsets
LetΦbeafamilyofballsinametricspace(M,ρ).
Definition3.6. LetE ⊂M,α>0,ε>0.Thentheα-dimensionalpackingpremea-
sureofaboundedsetE withrespecttoΦisdefinedby
Pα(E,Φ):=sup |E |α ,
ε i
(cid:26) i (cid:27)
X
where the supremum is taken over all uncentered ε-packings {E } ⊂ Φ of E (if
i
{E }=∅,thenPα(E,Φ)=0).
i ε
Definition3.7. Theα-dimensionalpackingquasi-measureofasetE w.r.t.Φisde-
finedby
Pα(E,Φ):= limPα(E,Φ).
0 ε
ε→0
Definition3.8. Theα-dimensionalpackingmeasurew.r.t.Φisdefinedby
Pα(E,Φ):=inf Pα(E ,Φ):E ⊂ E ,
0 j j
(cid:26) j (cid:27)
X [
wheretheinfimumistakenoverallatmostcountablecoverings{E }ofE,E ⊂M.
j j
PDP-transformations 379
Definition3.9. Thenonnegativenumber
dim (E,Φ):=inf α:Pα(E,Φ)=0
P
iscalledthepackingdimensionofasetE(cid:8)⊂M w.r.t.Φ. (cid:9)
Remark3.3. Inthedefinitionofdim (E,Φ),weuseduncenteredpacking.Butwe
P
willdenotethisdimensionwithoutindex(unc)because:
1. WewillworkinRn.Inthisspace,centeredanduncenteredpackingdimensions
areequal;
2. Thecenteredpackingdimensionw.r.t.somefamilyofballsisnotdefined.
Theorem3.2.
dimP(E,Φ)6dimP(unc)(E).
Proof. LetΦ bethefamilyofallopenballsofM.Then
0
Pα (E)=Pα(E,Φ ).
r(unc) r 0
SinceΦ⊆Φ ,bythesupremumpropertywehave
0
Pα(E,Φ)6Pα(E,Φ ).
r r 0
Bytheinequalityforpackingpremeasuresitfollowsthat
dimP(E,Φ)6dimP(unc)(E),
whichprovesthetheorem.
4 Faithfulnessoftheopenballsfamiliesforpackingdimensioncalculation
Definition4.1. SupposethatsomeopenballsfamilyΦsatisfiesthefollowingcondi-
tion:forallE ⊂M,dimP(unc)(E,Φ)=dimP(unc)(E).ThenΦissaidtobefaithful
foruncenteredpackingdimensioncalculation.
Remark4.1. ThenotionoffaithfulnessisintroducedfortheHausdorff–Besicovitch
dimensiondim [11].Itisclearthat
H
∀Φ⊂2M,dim (E,Φ)>dim (E).
H H
Theorem 4.1 (The sufficient condition for the open-ball family to be faithful for
packingdimensioncalculation). Supposethat
1. Φisafamilyofintervalsfrom[0;1];
2. ∃C >0:∀(a;b)⊂[0;1],∃∆(a;b)∈Φsuchthat:
(a) a+b ∈∆(a,b);
2
(b) ∆(a,b)⊂(a;b);
(c) b−a >C.
|∆(a,b)|
ThenΦisafaithfulopen-ballfamilyforpackingdimensioncalculation.
380 O.Slutskyi
Proof. Let E be anyset, α > 0, and r > 0. Let{E } = {(a ;b )} be a family of
i i i
disjointintervalssuchthat ai+bi ∈E andb −a <r.
2 i i
Thenthefollowinginequalityholds:
|E |α 6 ∆(a ,b ) α·Cα.
i i i
i i
X X(cid:12) (cid:12)
(cid:12) (cid:12)
Takingthesupremum(overallsetsofintervals{E }satisfyingthepreviouscon-
i
ditions),wehave
Pα(E)6 sup ∆(a ,b ) α·Cα.
r i i
{Ei}
(cid:12) (cid:12)
Anyset of intervals{∆(a ,b )} satis(cid:12)fies the co(cid:12)nditionsfrom the Pα (E,Φ)
i i r(unc)
definition.So,
sup ∆(a ,b ) α·Cα 6Pα (E,Φ)·Cα.
i i r(unc)
{Ei}
(cid:12) (cid:12)
Therefore, (cid:12) (cid:12)
Pα(E)6Pα (E,Φ)·Cα.
r r(unc)
Takingthelimitofbothsides,wehave
Pα(E)6Pα (E,Φ)·Cα.
0 0(unc)
TakingtheinfimumoverallpossiblecoveringsofthesetE,wehave
Pα(E)6Pα (E,Φ)·Cα
(unc)
and
dimP(E)6dimP(unc)(E,Φ).
Since[0;1]⊂R1,itfollowsthat
dimP(E)=dimP(unc)(E)
and
dimP(unc)(E)6dimP(unc)(E,Φ).
Using
dimP(unc)(E)>dimP(unc)(E,Φ) forallΦ,
weobtainthatΦisafaithfulopen-ballfamilyforthepackingdimensioncalculation.
˜
5 SufficientconditionsforQ-expansioncylindricintervalfamilytobefaithful
TheQ˜-expansionofrealnumbersisageneralizationofs-expansionandQ-expansion
andwasdescribed,forexample,in[4].
Theorem5.1. LetΦbethesystemofcylindricintervalsofsomeQ˜-expansion.Sup-
posethat
infq =q >0.
ij min
i,j
ThenΦisafaithfulballfamilyforpackingdimensioncalculation.