Some Random sequences related to average density of self-similar measures

Math. Nachr. 284, No. 11–12, 1567 – 1576 (2011) / DOI 10.1002/mana.200810183

Some Random sequences related to average densityof self-similar measures

Ying Xiong∗1 and Zhi-Xiong Wen∗∗2

1 School of Mathematical Sciences, South China University of Technology, Guangzhou, 510640, P. R. China2 Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, P. R. China

Received 11 August 2008, revised 14 January 2010, accepted 21 January 2010Published online 31 May 2011

Key words Weighted average, average density, law of iterated logarithm, self-similar measureMSC (2010) 28A80, 60G50

In this paper we study the limit behavior of weighted averages of some random sequence related to Bernoullirandom variables, and apply the results to average density of self-similar measures.

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction and main results

Let (Ω,P) be a probability space, and let {Xi}∞i=1 be a sequence of independent random variables with Bernoullidistribution P{Xi = 0} = 1 − p, and P{Xi = 1} = p, for i ∈ N, where p ∈ (0, 1) is a constant. Let λ > 1 be aconstant. For a sequence {an} of positive numbers, define a random sequence of weighted averages

AN =1∑N

n=1 an

N∑n=1

anλSn −np , N ≥ 1, (1.1)

where Sn = X1 + · · · + Xn .This paper is about the almost sure limit behavior of the sequence {AN }, which naturally arises in the study

of the average density of certain self-similar measures (see Section 2 for details). The fluctuation of the randomsequence {Sn −np} (as suggested by the law of the iterated logarithm) indicates that the sequence {AN } shouldbehave quite irregularly. This intuition will be justified in the sequel. We shall investigate this irregularity in somedetail and derive some consequences for the average density. We begin with the following notation

A∗ = lim supN →∞

AN , and A∗ = lim infN →∞

AN . (1.2)

Our first result is the following:

Theorem 1.1 Let {an} be a sequence of positive numbers such that∑∞

n=1 an = +∞. Then we have

A∗ = +∞, a.s. and either A∗ = +∞, a.s. or A∗ = 0, a.s.

As for the exact value of the lower limit A∗, we have some partial results under the regular variation condi-tions.

Theorem 1.2 Suppose that {log an} varies regularly with index ρ, i.e., an = exp(nρ�n

), where

{�n

}varies

slowly (see Definition 3.1), then we have

A∗ = lim infN →∞

AN =

{+∞, a.s. if 0 ≤ ρ < 1/2;

0, a.s. if ρ > 1/2.

∗ Corresponding author: e-mail: [email protected]∗∗ e-mail: [email protected]

c© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1568 Y. Xiong and Z.-X. Wen: Random sequence related to average density

For ρ = 1/2, i.e., an = exp(√

n�n

), we have


AN =

⎧⎪⎪⎨⎪⎪⎩

+∞, a.s. if �n ↓ 0 and∞∑

n=1

�n/n < +∞;

0, a.s. if lim supn→∞

√log log n/�n < +∞.

In Theorems 1.1 and 1.2, the condition∑∞

n=1 an = +∞ is assumed. Now we consider the case of∑∞n=1 an < +∞. In this case, it is equivalent to studying the sums

∑∞n=1 anλSn −np . We have the following

Theorem 1.3 Let {an} be a sequence of positive numbers. Suppose that there exists a constant β > 1 suchthat

∞∑n=1

anβ√

n = +∞,

then we have

∞∑n=1

anλSn −np = +∞, a.s.

Remark 1.4 This is nearly sharp in the following sense: given a sequence of nonnegative numbers {bn}satisfying lim supn→∞ bn = +∞, there exists a positive sequence {an} such that

∞∑n=1

anβbn√

n = +∞, but∞∑

n=1

anλSn −np < +∞, a.s. (1.3)

As a corollary of Theorem 1.3, we have

Corollary 1.5 Suppose that the sequence of positive numbers {an} satisfy

∞∑n=1

anβ√

n = +∞, but∞∑

n=1

an < +∞,

where β > 1 is a constant. Then A∗ = A∗ = +∞, a.s.

As applications, in Section 2, using the idea of average density (see Bedford and Fisher [1]), we consider thelocal geometric structure of self-similar measures with support on the Cantor set C. Section 3 is devoted to somepreliminaries. The proofs of the theorems are postponed to Section 4.

2 Average density

We first recall some results about density theory of general Radon measures (see [8]). Let μ be a Radon measureon an Euclidean space and s > 0, the s-lower and upper densities of μ at the point x are defined respectively by

Dsμ(x) := lim inf

r→∞

μ(B(x, r)

)rs

, and Ds

μ(x) := lim supr→∞

μ(B(x, r)

)rs

. (2.1)

In case that Dsμ(x) = D

s

μ(x), the common value will be denoted by Dsμ(x) which is called the s-density of the

Radon measure μ at the point x. Existence of the s-density is a rare phenomenon as shown by Marstrand [7].

Theorem 2.1 (Marstrand) Let s be a positive number. Suppose that there exists a Radon measure μ on Rn

such that the density Dsμ(x) exists and is positive and finite in a set of positive μ measure. Then s is an integer.

So if s is not an integer, then the ratio μ(B(x, r)

)/rs “oscillates” more or less between Ds

μ(x) and Dsμ(x). It

is appropriate to use a form of averaging that describe this oscillation. Indeed, in the following theorem, Bedfordand Fisher proved in [1] that a version of the average density does exist for a special class of measures.

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Math. Nachr. 284, No. 11–12 (2011) / www.mn-journal.com 1569

Theorem 2.2 Let C be the middle-third Cantor set, and let s = dimH C = log 2/ log 3. The order-two-density(in dimension s) for Hausdorff measure Hs

As2(x) := lim

δ→0

1∫ 1δ

r−1dr

∫ 1

δ

Hs(B(x, r)

)rs

r−1dr

exists and is a constant for Hs almost all x ∈ C.

Actually, by the results of Falconer [3] and Patzschke and M. Zahle [9] this holds more generally for self-similar and even self-conformal sets in the Euclidean space.

The results of Bedford and Fisher only concern a special self-similar measure on C, namely, the s-dimensionHausdorff measure. It is therefore natural to ask: what will happen to other measures? We shall consider the classof self-similar measures which are defined as follows (see J. E. Hutchinson [6]).

Definition 2.3 (Self-similar measure) A measure μ on the Cantor set C is said to be self-similar if there existsa number p ∈ [0, 1] such that

μ = (1 − p)μ ◦ S−10 + pμ ◦ S−1

1 , (2.2)

where S0 and S1 are the similitudes given by S0 : x → x/3 and S1 : x → (x + 2)/3.

It is proved in J. E. Hutchinson [6] that there exists a unique probability measure satisfying (2.2). This uniqueprobability measure will be denoted by μp in the sequel. We shall consider a more general type of average densityof μp .

Definition 2.4 (General average density) Let s > 0 and let f : (0, 1] → (0,+∞) be a measurable functionsatisfying ∫ δ

0f(r) dr = +∞, and

∫ 1

δ

f(r) dr < +∞ (2.3)

for all 0 < δ < 1. The (s, f)-lower and upper average densities of μp at x are defined by

Asf (x) := lim inf

δ→0As

f (x, δ), and As

f (x) := lim supδ→0

Asf (x, δ),

where

Asf (x, δ) =

1∫ 1δ

f(r) dr

∫ 1

δ

μp

(B(x, r)

)rs

f(r) dr.

In case that Asf (x) = A

s

f (x), the common value will be denoted by Asf (x) which is called the (s, f)-average

density of the self-similar measure μp at the point x.

Remark 2.5 The local dimension of μp is given by (cf. [4, Proposition 10.2])

dimloc μp = limr→0

log μp

(B(x, r)

)log r

=−p log p − (1 − p) log(1 − p)

log 3, μp -a.e.

Therefore, if s < dimloc μp (resp. > dimloc μp), then Asf = 0 (resp. = +∞) for μp almost all x ∈ C. So in the

sequel we always assume that

s = dimloc μp =−p log p − (1 − p) log(1 − p)

log 3. (2.4)

To investigate the average density Asf , it is convenient to make use of the symbolic representation of the

Cantor sets which we recall briefly now. Let {0, 1}N be the space of all sequences of 0’s and 1’s, let p ∈ (0, 1),let p0 = 1 − p and let p1 = p. We may define a Bernoulli measure νp on {0, 1}N by

νp

([x1 , . . . , xn ]

)=

n∏i=1

pxi,

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where [x1 , . . . , xn ] :={y ∈ {0, 1}N : y1 = x1 , . . . , yn = xn

}. It is known that the measure spaces

({0, 1}N, νp

)and (C, μp) are conjugate (see [11, p. 54]) via the map

(x1 , x2 , . . . ) −→∑i≥1

2xi · 3−i .

We can infer from the conjugacy that

(i) The sequence of random variables X1 ,X2 , . . . defined on (C, μp) by Xi(x) = xi , for x = 2∑∞

i=1 3−ixi ,is a sequence of independent random variables with identical distribution

P{Xi = 0} = p0 = 1 − p, and P{Xi = 1} = p1 = p, for i ∈ N.

(ii) For x = 2∑∞

i=1 3−ixi , and 3−n ≤ r < 3−n+1 ,

3(n−1)sn∏

i=1

pxi≤

μp

(B(x, r)

)rs

< 3nsn−1∏i=1

pxi. (2.5)

In the sequel, let us agree that g(x) ≈ h(x) means that there exists a constant c > 1 (independent of x) suchthat

c−1h(x) ≤ g(x) ≤ ch(x). (2.6)

By (2.4) and (2.5), We have

μp

(B(x, r)

)rs

≈ 3nsn∏

i=1

pxi=

(p

1 − p

)Sn(x)−np

, (2.7)

where Sn (x) =∑n

i Xi(x) = x1 + · · ·+xn . Fix δ > 0, let N be determined by 3−N ≤ δ < 3−N +1 , then rewritethe integral ∫ 1

δ

μp

(B(x, r)

)rs

f(r) dr =∫ 3−N + 1

δ

μp

(B(x, r)

)rs

f(r) dr +N −1∑n=1

∫ 3−n + 1

3−n

μp

(B(x, r)

)rs

f(r) dr.

Putting an =∫ 3−n + 1

3−n f(r) dr, λ = p/(1 − p) and applying (2.7), we have

∫ 1

δ

μp

(B(x, r)

)rs

f(r) dr ≈ t(δ)aN λSN (x)−N p +N −1∑n=1

anλSn (x)−np ,

where

0 ≤ t(δ) =

∫ 3−N + 1

δf(r) dr∫ 3−N + 1

3−N f(r) dr≤ 1.

So we find that

Asf (x, δ) ≈ t(δ)aN λSN (x)−N p +

∑N −1n=1 anλSn (x)−np

t(δ)aN +∑N −1

n=1 an

.

Since the function h(t) = a+ctb+dt with a, b, c, d > 0, is monotone on [0,1], by (1.1) and (2.6), there exists a

constant c > 1 which is independent of x and δ such that

c−1 min(AN −1(x), AN (x)

)≤ As

f (x, δ) ≤ cmax(AN −1(x), AN (x)

).

Consequently, we obtain

c−2A∗(x) ≤ c−1Asf (x) ≤ A∗(x) ≤ A∗(x) ≤ cA

s

f (x) ≤ c2A∗(x), (2.8)

where A∗ and A∗ are defined in (1.2). Therefore the average density of μp relates to the random sequence {AN }.

Without loss of generality, we assume that p > 1/2 in what follows, then λ = p1−p > 1. Since

∫ 10 f(r) dr =

+∞, it follows that∑∞

n=1 an = +∞. By Theorem 1.1 and inequality (2.8), we have



Theorem 2.6 Let f > 0 be a measurable function defined on (0, 1] such that (2.3) holds, then

As

f (x) = +∞, μp -a.e., and either Asf (x) = +∞, μp -a.e., or As

f (x) = 0, μp -a.e.

Furthermore, we can apply Theorem 1.2 and inequality (2.8) to get

Theorem 2.7 Let f > 0 be a measurable function defined on (0, 1] such that (2.3) holds, and

log an = log(∫ 3−n + 1

3−n

f(r) dr

)

varies regularly with index ρ (see Definition 3.1), i.e.,

an = exp(nρ�n

), (2.9)

where {�n} varies slowly. Then

Asf (x) =

{+∞, μp -a.e. if 0 ≤ ρ < 1/2;

0, μp -a.e. if ρ > 1/2.

For ρ = 1/2, i.e., an = exp(√

n�n

), we have

Asf (x) =

⎧⎪⎪⎨⎪⎪⎩

+∞, μp -a.e. if �n ↓ 0 and∞∑

n=1

�n/n < +∞;

0, μp -a.e. if lim supn→∞

√log log n/�n < +∞.

To conclude this section, we give some examples.

Example 2.8 Let f(r) = r−1 , then the (s, f)-average density become the order-two density in Bedfordand Fisher [1]. In this case,

an =∫ 3−n + 1

3−n

f(r) dr = log 3.

It follows that ρ = 0 in (2.9), therefore by Theorems 2.6 and 2.7, Asf = A

s

f = +∞, μp -a.e.

Example 2.9 Let f(r) = r−1−ε , where ε > 0, then

an =∫ 3−n + 1

3−n

f(r) dr = ε−1(3ε − 1) · 3ε(n−1) .

It follows that ρ = 1 in (2.9), therefore by Theorems 2.6 and 2.7, Asf = 0, μp -a.e., and A

s

f = +∞, μp -a.e.

3 Preliminaries

This section contains preliminaries that are needed in the proofs of the main theorems.

3.1 Regular variation

We recall some basic facts of regular variation needed in this paper. For more details, see Bingham, Goldieand Teugels [2, Chapter 1].

Definition 3.1 (Regular variation) A positive sequence {�n} is said to be slowly varying if

limn→∞

�[γn ]

�n= 1, for all γ > 0.

A positive sequence {n} is called regularly varying of index ρ if

n = nρ�n ,

where {�n} is a slowly varying sequence and ρ ∈ R.



We cite a theorem about slowly varying sequence, see [2, p. 58 and p. 16].

Theorem 3.2 A positive sequence {�n} varies slowly if and only if for every ρ > 0,

max0≤k≤n

{kρ�k

}∼ nρ�n , and max

k≥n

{k−ρ�k

}∼ n−ρ�n ,

where gn ∼ hn means gn/hn → 1 as n → ∞. Moreover, if {�n} varies slowly, then

limn→∞

nρ�n =

{+∞, if ρ > 0;

0, if ρ < 0.

3.2 Theorems in probability theory

First we recall a strong law for the maximum cumulative sum of independent random variables due to W. M.Hirsch [5].

Theorem 3.3 (Hirsch) Let X1 , . . . , Xn be independent random variables with identical distribution andEXi = 0, EX2

i < +∞, E|Xi |3 < +∞, for 1 ≤ i ≤ n. Write Mn = max1≤k≤n Sk , where Sk = X1 + · · ·+ Xk ,then for arbitrary αn ↓ 0,

P{Mn < αn

√n i.o.

}=

⎧⎪⎨⎪⎩

1 if∞∑

n=1

αn

n= +∞;

0 otherwise.

Then we list two lemmas involving the law of iterated logarithm. This two lemmas follow by the law of iteratedlogarithm and the central limit theorem. We skip the proofs since the argument is routine.

Lemma 3.4 Let X1 ,X2 , . . . be a sequence of independent random variables with Bernoulli distributionP(Xk = 1) = p, P(Xk = 0) = 1 − p for all k ≥ 1, where p ∈ (0, 1). Suppose that {ni} is a strictlyincreasing sequence of positive integers. For any α > 0, let

E(ni) ={Sni

− nip > α√

ni

},

where Sni=

∑ni

k=1 Xk . Then with probability one, the events E(ni) occur infinitely many times.

Lemma 3.5 Let X1 ,X2 , . . . be a sequence of independent identically distributed random variables withEXi = 0 and |Xi | < 1 for all i ≥ 1, put Sn =

∑ni=1 Xi , sn =

√Var Sn , φ(n) =

√2 log log n. Then for

any γ > 1, with probability one, the events

Eγ (n) ={

maxn<j≤γn

Sj < −αφ(n)sn

}

occur infinitely often or only finitely often according as α < 1 or α > 1.

4 Proofs of the main results

4.1 Proof of Theorem 1.1

To show that A∗ = +∞, a.s., we need to consider two cases separately.CASE (I) If supn{an} = +∞, we can find a subsequence {ni} such that max1≤k≤ni

ak = ani, and so

A∗ ≥ lim supi→∞

Ani≥ lim sup

i→∞

aniλSn i

−ni p

niani

= lim supi→∞

λSn i−ni p

ni.

Taking α = 1 in Lemma 3.4, we have


λSn i−ni p

ni≥ lim sup

i→∞

λ√

ni

ni= +∞, a.s.



CASE (II) If supn{an} = M < +∞, since∑∞

n=1 an = +∞, there exists a subsequence {ni} such thatani

≥ n−2i , and so


Ani≥ lim sup

i→∞

aniλSn i

−ni p

anin2

i · Mni= lim sup

i→∞

λSn i−ni p

Mn3i

.

In Lemma 3.4, taking α = 1, then we have


λSn i−ni p

Mn3i

≥ lim supi→∞

λ√

ni

Mn3i

= +∞, a.s.

Therefore, in all cases, we have proved A∗ = +∞, a.s.Now we consider the lower limit A∗. The result follows from the fact that the probability of a symmetric

event is only 0 or 1 (for this fact and the definition of symmetric event, see [10, p. 382]). Let us first show that{A∗ ∈ B} is a symmetric event for every Borel subset B of R. Let π be a finite permutation and n′ > 0 be suchthat π(n) = n for all n > n′. Since

∑∞n=1 an = +∞,


1∑Nn=1 an

N∑n=1

anλSn −np = lim infN →∞

1∑Nn=1 an

N∑n=n ′+1

anλSn −np

= lim infN →∞

1∑Nn=1 an

N∑n=n ′+1

anλXπ ( 1 ) +···+Xπ (n )−np ,

thus we see that {A∗ ∈ B} is symmetric.We are going to prove that either A∗ = +∞, a.s. or A∗ = 0, a.s. by reduction to absurdity. Assume, otherwise,

that P{A∗ ∈ (0,+∞)

}= 1. Since the events

{A∗ ∈ [λk , λk+1)

}are all symmetric for any k ∈ Z, there exists

an unique k ∈ Z such that P{A∗ ∈ [λk , λk+1)

}= 1. Let D denote the event that{

lim infN →∞

1∑Nn=1 an

N∑n=1

anλSn −X 1 −np ∈ [λk , λk+1)

},

then the event D is independent of the random variable X1 . Therefore

P(D|{X1 = 1}

)= P(D) = P

(D|{X1 = 0}

)= 1,

hence

P(D ∩ {X1 = 1}

)= P{X1 = 1} = p.

But D ∩ {X1 = 1} ⊂{A∗ ∈ [λk+1 , λk+2)

}is a set of probability zero. This contradiction establishes our

result. �


Put Mn = maxk≤n (Sk − kp), and M∗n = maxk≤n log ak . We split the proof into two cases.

CASE (I) If either {log an} varies regularly with index ρ < 1/2, or log an =√

n�n , where {�n} is a slowlyvarying sequence satisfying �n ↓ 0 and

∑∞n=1 �n/n < +∞.

By Theorems 3.2 and 3.3, we see that, with probability one, eventually, Mn log λ > M∗n + 2 log n. Therefore,

A∗ > lim infn→∞

λMn

neM ∗n

> lim infn→∞

n2eM ∗n

neM ∗n

= +∞, a.s.

CASE (II) If either {log an} varies regularly with index ρ > 1/2, or log an =√

n�n , then {�n} is a slowlyvarying sequence satisfying

lim supn→∞

√log log n/�n < +∞. (4.1)



It follows from the law of iterated logarithm that

lim supn→∞

Mn√2p(1 − p)n log log n

= 1, a.s.

By (4.1), we know that

lim supn→∞

Mn

M∗n

< +∞, a.s.

According to Theorem 3.2, for any positive integer γ, we have

limn→∞

M∗n

M∗γn

= γ−ρ .

Hence we can take γ sufficiently large such that, with probability one, eventually,

2 log n + M∗n + Mn log λ < M∗

γn . (4.2)

Now by Lemma 3.5, for almost all ω ∈ Ω, there exists a subsequence {ni} depending on ω such that

maxni <k≤γni

(Sk (ω) − kp

)< −

√p(1 − p)n log log n.

This result suggests to rewrite the sum

γni∑k=1

akλSk(ω )−kp =ni∑

k=1

akλSk(ω )−kp +γni∑

k=ni +1

akλSk(ω )−kp .

It is obvious that

∑γni

k=ni +1 akλSk(ω )−kp∑γni

k=1 ak→ 0, as i → ∞.

Since (4.2) holds with probability one, we can assume that ω also satisfies (4.2). So eventually,

∑ni

k=1 akλSk(ω )−kp∑γni

k=1 ak<

nieM ∗

n i λMn i(ω )

eM ∗γ n i

<1ni

.

Therefore, we have shown that if (4.1) holds, A∗ = 0, a.s. �


Since∑∞

n=1 anβ√

n = +∞, there exists a subsequence {ni} such that ani≥ n−2

i β−√ni . Now we apply

Lemma 3.4. Taking α = log 2β/ log λ, for almost all ω ∈ Ω, we can find a subsequence {mj} of {ni}, de-pending on ω, such that λSm j

(ω )−mj p > (2β)√

mj . Therefore,

∞∑n=1

anλSn(ω )−np ≥∞∑

j=1

amj(2β)

√mj ≥

∞∑j=1

m−2j 2

√mj = +∞.

We conclude that∑∞

n=1 anλSn −np = +∞, a.s.



4.4 Demonstration of Remark 1.4

Given a sequence of nonnegative numbers {bn} satisfying lim supn→∞ bn = +∞, we shall construct a sequence{an} such that (1.3) holds.

By the inequality for the probability of large deviations (see [10, p. 69]),

P

{Sn

n− p ≥ ε

}≤ e−2nε2

,

we obtain

P

{Sn − np ≥ bn log β

2 log λ

√n

}≤ exp

(−b2

n log2 β

2 log2 λ

).

Since lim supn→∞ bn = +∞, there exists a sequence {ni} such that

∞∑i=1

exp

(−

b2ni

log2 β

2 log2 λ

)< +∞.

Applying Borel-Cantelli lemma, we have

P

{Sni

−nip ≥ bnilog β

2 log λ

√ni, i.o. for i

}= 0. (4.3)

Now we put

an =

{β−bn

√n , if n ∈ {ni};

β−n , otherwise.(4.4)

According to the law of the iterated logarithm, we have

P

{Sn − np ≥ n log β

2 log λ, i.o. for n

}= 0. (4.5)

It follows from (4.3)–(4.5) that

∞∑n=1

anλSn −np < +∞, a.s., but∞∑

n=1

anβbn√

n = +∞.

Acknowledgements The authors would like to thank Professor Jihua Ma for suggesting this problem and helpful discus-sions. This work was supported by the National Natural Science Foundation of China (Grant No. 10771164, 11071082,11071090). The first author now also supported by Fundamental Research Funds for the Central Universities, SCUT(D2104900) and Research Fund for the Doctoral Program of Higher Education of China (20100172120027).

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Some Random sequences related to average density of self-similar measures