Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II

ISSN:1083-589X in PROBABILITY

Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II

David Nualart

Jason Swanson

Abstract

The purpose of this paper is to provide a complete description the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brow- nian motion with Hurst parameterH = 1/6.

Keywords:fractional Brownian motion; cubic variation; convergence in law.

AMS MSC 2010:Primary 60G22, Secondary 60F17.

Submitted to ECP on May 30, 2013, final version accepted on August 17, 2013.

1 Introduction

Suppose thatB ={B(t), t≥0}is a fractional Brownian motion with Hurst parameter H = ¹₆. Letbxcdenote the greatest integer less than or equal tox. In [6], Nualart and Ortiz-Latorre proved that the sequence of sums,

Wn(t) =

bntc

X

j=1

(B(j/n)−B((j−1)/n))³, (1.1)

converges in law to a Brownian motionW ={W(t), t≥0}, with varianceκ²tgiven by κ²=3

4 X

m∈Z

(|m+ 1|^1/3+|m−1|^1/3−2|m|^1/3)³.

The processW is related to the signed cubic variation ofB. A detailed analysis of this process has been recently developed by Swanson in [8], considering this variation as a class of sequences of processes.

In [1], Burdzy, Nualart and Swanson studied the convergence in distribution of the sequence of two-dimensional processes{(Wa_n(t), Wb_n(t))}, where{an}^∞_n=1 and{bn}^∞_n=1 are two strictly increasing sequences of natural numbers converging to infinity. A basic assumption for the results of [1] and also for the results of this paper is that L_n → L ∈[0,∞], where Ln =bn/an. By [1, Corollary 3.6], ifL∈ {0,∞}, thenWa_n and Wb_n

converge to independent Brownian motions. We will therefore assume thatL∈(0,∞). The functionfL(x) =P

m∈Zfm,L(x), where

fm,L(x) = (|x−m+ 1|^1/3+|x−m−L|^1/3− |x−m|^1/3− |x−m+ 1−L|^1/3)³, (1.2)

∗David Nualart is partially supported by grant DMS-1208625 from the NSF.

†University of Kansas, USA. E-mail:[email protected]

‡University of Central Florida, USA. E-mail:[email protected]

plays a fundamental role in the analysis of the convergence of {(Wan, W_bn)}. Under some conditions, the limit in distribution of this sequence is a two-dimensional Gaussian processX^ρ, independent ofB, whose components are Brownian motions with variance κ²t, and with covarianceRt

0ρ(s)dsfor some function ρ. In terms of the function ρ ∈ C[0,∞), the processX^ρcan be expressed as

X^ρ(t) = Z t

0

σ(s)dW(s), (1.3)

whereσis given by

σ(t) =κ

p1− |κ⁻²ρ(t)|² κ⁻²ρ(t)

0 1

, (1.4)

and W = (W¹, W²) is a standard, 2-dimensional Brownian motion. More specifically, the main result of [1] is the following theorem, which is obtained using the central limit theorem for multiple stochastic integrals proved by Peccati and Tudor in [7] (see also [3]).

Theorem 1.1. LetI={n:L_n =L}andc_n= gcd(a_n, b_n). Then(B, W_an, W_bn)⇒(B, X^ρ) in the Skorohod spaceD_R3[0,∞)asn→ ∞, in the following cases:

(i) The setI^c is finite (which impliesL∈Q). In this case, ifL =p/q, wherep, q ∈N are relatively prime, then for allt≥0,

ρ(t) = 3 4p

q

X

j=1

fL(j/q).

(ii) There existsk∈N^{such that}bn=k modanfor alln. In this case, for allt≥0, ρ(t) = 3

4Lf_L(kt).

(iii) The setIis finite andcn→ ∞. In this case, for allt≥0,

ρ(t) = 3 4L

Z 1

0

f_L(x)dx.

This type of result was motivated by the relationship between higher signed varia- tions of fractional Brownian motions and the change of variable formulas in distribution for stochastic integrals with respect to these processes that have appeared recently in the literature (see [2, 4, 5]).

Theorem 1.1 covers many simple and interesting pairs of sequences, and helps to tell a surprising story about the asymptotic correlation between the sequences,{Wa_n} and{Wbn}, both of which are converging to a Brownian motion. For example, by The- orem 1.1(i), we may conclude that the asymptotic correlation ofW_n(t) and W_2n(t) is a constant that does not depend on t, and whose numerical value is approximately 0.201. Likewise, Theorem 1.1(iii) shows that the asymptotic correlation ofW_n2(t)and W_n(n+1)(t) is not dependent ont and is approximately 0.102. Perhaps more surpris- ingly, Theorem 1.1(ii) shows that the asymptotic correlation ofWn(t)andWn+1(t)does depend ont. Numerical calculations suggest that the correlation varies greatly witht, converging to 1 ast↓0, and being as low as about 0.075 fort= 0.8.

Nonetheless, there are many simple and interesting pairs of sequences that arenot covered by Theorem 1.1. For example, the sequencesan =n²andbn = (n+ 1)²are not covered; nor are the sequencesan = 2nandbn= 3n+ 1. Additionally, many sequences whose ratios converge to an irrational number are not covered by this theorem.

The purpose of this paper is to provide a complete description of the asymptotic behavior ofW_an(t)andW_bn(t)for all sequences{a_n} and{b_n}. We will show that the asymptotic correlation depends only onL= limLn whenLis irrational; and whenLis rational, it depends also onliman|Ln−L|. In the next section we state and prove this result and provide some remarks and examples.

We should remark that, as discussed at the end of the introduction to the previous paper [1], it is natural to consider generalizations of (1.1), such as

W_n(t) = 1

√n

bntc

X

j=1

f(n^H(B(j/n)−B((j−1)/n))),

wheref : R → R is square integrable with respect to the standard normal distribu- tion and has an expansion into a series of Hermite polynomials of the form f(x) = P∞

k=qakHk(x), withq ≥1, andB is fractional Brownian motion with Hurst parameter H <1/(2q). In this case, the computation of the asymptotic correlations,

n→∞lim E[Wa_n(t)Wb_n(t)],

appears to be rather involved and requires methods and ideas beyond those developed in this and the previous paper, even in the particular caseH = 1/(2k)andf(x) =x^k for oddk. It seems plausible that our results have natural extensions to the more general setting, but the study of asymptotic correlations with more general H and f will be reserved for future papers.

2 Main result

Let X^ρ the two-dimensional process defined in (1.3). Recall thatfL =P

m∈Zfm,L, wherefm,L is the function defined in (1.2). By [1, Lemma 2.6], the series defining fL

converges uniformly on[0,1]. Also note thatfL is periodic with period 1. We first need the following technical result.

Lemma 2.1. LetL=p/q, wherep, q ∈N are relatively prime numbers. Then, for any x∈R^andη= 1, . . . , qwe havefL(ηL−x) =fL(ηLe +x), whereeη=q−η+ 1.

Proof. For anym∈Z^setme =−m+ 1 +p. Then

fm,L(ηL−x) =

ηp

q −x−m+ 1

1/3

+

ηp

q −x−m−p q

1/3

−

ηp

q −x−m

1/3

−

ηp

q −x−m+ 1−p q

1/3!³

=

−ηp

q +x−me +p

1/3

+

−ηp

q +x−me + 1 +p+p q

1/3

−

−ηp

q +x−me + 1 +p

1/3

−

−ηp

q +x−me +p+p q

1/3!3

.

Notice thatηLe =p+^p_q −^ηp_q . Therefore,

f_m,L(ηL−x) = |ηLe +x−me −L|^1/3+|eηL+x−me + 1|^1/3

− |eηL+x−me −L+ 1|^1/3− |eηL+x−m|e ^1/33

=f

m,Le (eηL+x).

As a consequence, fL(ηL−x) = X

m∈Z

fm,L(ηL−x) = X

m∈Z

fm,Le (ηLe +x) = X

m∈Ze

fm,Le (ηLe +x) =fL(ηLe +x),

which completes the proof.

The next result is the main theorem of this paper. Together with the casesL= 0and L =∞, covered in [1, Corollary 3.6], this theorem gives a complete description of all subsequential limits of(Wa_n, Wb_n)for any pair of subsequences of{Wn}.

Theorem 2.2. Let{an}^∞_n=1and{bn}^∞_n=1be strictly increasing sequences inN^{. Let}Ln= b_n/a_nand supposeL_n →L∈(0,∞). Letδ_n=L_n−L. Then,(B, W_an, W_bn)⇒(B, X^ρ)in D_R3[0,∞)asn→ ∞, in the following cases:

(i) L∈Q^andan|δn| →k∈[0,∞). In this case, if we writeL=p/q, wherep, q∈N^are relatively prime, then, for allt≥0,

ρ(t) = 3 4p

q

X

j=1

f_L j

q+kt

.

(ii) L∈Q^andan|δn| → ∞, orL /∈Q. In this case, for allt≥0,

ρ(t) = 3 4L

Z 1

0

fL(x)dx.

Note that between the two parts of this theorem, there is, at least formally, a sort of continuity ink. For fixedq, we have

Z t

0

3 4p

q

X

j=1

f_L j

q +ks ds→ Z t

0

3 4L

Z 1

0

f_L(x)dx

ds,

ask→ ∞. In fact, sincefLis periodic with period 1, for anyj = 1, . . . , q, Z t

0

f_L j

q +ks

ds= 1 k

Z kt

0

f_L j

q+x

dx

= 1 k

Z kt−bktc

0

f_L j

q +x

dx+bktc k

Z 1

0

f_L(x)dx→t Z 1

0

f_L(x)dx,

ask→ ∞.

To elaborate on the conditions in the two parts of this theorem and their connections to Theorem 1.1, first note that ifLis rational andLn6=L, then

a_n|δ_n|=

b_nq−a_np q

≥ 1

q, (2.1)

since the numerator is a nonzero integer. It follows that whenL∈Q^{, we have}an|δn| →0 if and only ifLn =Lfor all but finitely manyn. Therefore, Theorem 2.2(i) withk= 0is equivalent to Theorem 1.1(i).

Next, ifL ∈Q^,L_n 6=L for all but finitely manyn, andc_n = gcd(a_n, b_n)→ ∞, then (2.1) shows that fornsufficiently large,a_n|δ_n| ≥c_n/q→ ∞. Hence, Theorem 1.1(iii) is a special case of Theorem 2.2(ii).

Lastly, to see that Theorem 1.1(ii) is a special case of Theorem 2.2(i), suppose there existsk∈N ^{such that}bn =k modan for alln. Thenbn =νnan+kfor some integers

ν_n. Thus,L_n =ν_n+k/a_n. Lettingn→ ∞we see that the sequenceν_n must converge toL. Taking in to account that theν_n’s are integers, this implies thatL∈N^andν_n=L for all but finitely manyn. We therefore havean|δn|=|bn−anL|=k, for large enough n. In this case, usingp=Landq= 1and the fact thatfLis periodic with period 1, we find that the functionρin Theorem 2.2(i) agrees with the functionρin Theorem 1.1(ii).

Before giving the formal proof of Theorem 2.2 we would like to explain the main ideas in comparison with the proof of Theorem 1.1. We denote by {x} = x− bxcthe fractional part ofx(not to be confused with the set whose unique element isx). In [1], it is shown that the covariance between the components of the limit processX^ρis given by

Cov (X₁^ρ, X₂^ρ) = Z t

0

ρ(s)ds= 3 4L

X

m∈Z n→∞lim

1 an

bantc

X

j=1

fm,L({jLn}),

provided the above limits exist for each m ∈ Z. The principal challenge in analyzing these limits has been that the above summands, fm,L({jLn}), could not be replaced by f_m,L({jL}). This is because, although L_n is close to L for large n, {jLn} is not uniformly close to{jL} asj ranges from 1 toba_ntc. In [1], we studied these limits via the decomposition

bantc

X

j=1

fm,L({jLn}) =αn qn−1

X

j=0

fm,L({jLn}) +

r_n

X

j=1

fm,L({jLn}).

Here,Ln=bn/an=pn/qn, wherepn andqnare relatively prime, andbantc=αnqn+rn

withα_n∈Z^and0≤r_n< q_n.

To prove Theorem 2.2 in the case thatL∈Q, we use a different decomposition. Let L=p/q, wherep, q∈Nare relatively prime. We then write

bantc

X

j=1

f_m,L({jL_n})≈

q

X

η=1 α_n−1

X

i=0

f_m,L({(iq+η)L_n}).

In this case, sinceqis fixed and finite, we are able to use the approximation

bantc

X

j=1

fm,L({jLn})≈

q

X

η=1 α_n−1

X

i=0

fm,L({iqLn+ηL}).

SinceqL=p, we haveiqL_n=ip+iqδ_n. Thus, we have

bantc

X

j=1

fm,L({jLn})≈

q

X

η=1 αn−1

X

i=0

fm,L({iqδn+ηL}).

Using a Riemann-sum argument, we will show that for each fixedη,

α_n−1

X

i=0

fm,L({iqδn+ηL})≈ 1 qδ_n

Z anδnt

0

fm,L({x+ηL})dx,

giving

Z t

0

ρ(s)ds= 3 4L

X

m∈Z

1 q

q

X

η=1 n→∞lim

1 anδn

Z anδnt

0

f_m,L({x+ηL})dx.

We then prove the theorem case-by-case, depending on the asymptotic behavior of the sequence anδn. Note that the actual analysis in the proof is made somewhat more delicate by the fact thatδnmay be negative.

For the caseL /∈ Q, the proof will be done by adapting the method of proof of the equidistribution theorem based on Fourier series expansions. This theorem says that for any intervalI⊂[0,1),

n→∞lim 1

n|{k:{kL} ∈I,1≤k≤n}|=|I|, and a simple proof can be found in [9, Theorem 1.8].

Proof of Theorem 2.2. Let

Sn(t) =E[Wa_n(t)Wb_n(t)].

In order to show that(B, Wa_n, Wb_n) ⇒ (B, X^ρ) in D_R3[0,∞), by [1, Theorem 3.1 and Lemma 3.5], it will suffice to show that

S_n(t)→ Z t

0

ρ(s)ds, (2.2)

for eacht≥0.

Fixt≥0. SinceWn(t) = 0ifbntc= 0, we may assumet >0andnis sufficiently large so thatbantc>0andbbntc>0. Recall that{x}=x− bxc, and letfb_m,L(x) =f_m,L({x}), wheref_m,Lis the function introduced in(1.2).

In the reference [1] it is proved (see [1, (3.18), (3.20), and Remark 3.3]) that

n→∞lim S_n(t) = 3 4L

X

m∈Z

n→∞lim β(m, n),e (2.3)

where

β(m, n) =e 1 an

bantc

X

j=1

fb_m,L(jL_n),

provided that, for each fixedm∈Z, the limitlim_n→∞β(m, n)e exists. The proof will now be done in several steps.

Step 1. AssumeL ∈ Q^and an|δn| →k ∈(0,∞]. Let us writeL =p/q, where pand q are relatively prime. Choosen0 such that for alln≥n0, we havebantc> q. For each n≥n₀, writebantc=α_nq+r_n, whereα_n ∈N^and0≤r_n < q. Sincea_n→ ∞andfb_m,L is bounded, it follows that

n→∞lim β(m, n) = lime

n→∞

1 an

α_nq

X

j=1

fb_m,L(jL_n)

= lim

n→∞

1 an

q

X

η=1 αn−1

X

i=0

fbm,L((iq+η)Ln)

= lim

n→∞

1 an

q

X

η=1 αn−1

X

i=0

fbm,L(ip+ηL+ (iq+η)δn)

=

q

X

η=1

n→∞lim 1 an

α_n−1

X

i=0

fbm,L(ηL+ sgn(δn)xi)

,

wherexi = (iq+η)|δn|. Our assumption that an|δn| → k ∈ (0,∞] implies that there existsn1≥n0such thatδn6= 0for alln≥n1. Set∆x=xi+1−xi=q|δn|. Then

n→∞lim β(m, n) =e 1 q

q

X

η=1

n→∞lim 1 a_n|δn|

α_n−1

X

i=0

fbm,L(ηL+ sgn(δn)xi)∆x

.

Letε >0be arbitrary. Sincef_m,Lis uniformly continuous on[0,1], we may findn₂≥n₁ such that for alln≥n₂,

sup

|x−y|≤∆x x,y∈[0,1]

|fm,L(x)−fm,L(y)|< ε.

Note that ifbxc=byc, then{x} − {y}=x−y. Thus, sup

|x−y|≤∆x bxc=byc

|fb_m,L(x)−fb_m,L(y)|< ε, (2.4)

for alln≥n2. Let

Jn={0≤i < αn :bηL+ sgn(δn)xic=bηL+ sgn(δn)xi+1c}.

Note that ifi∈Jn andx∈[xi, xi+1], then

bηL+ sgn(δn)xc=bηL+ sgn(δn)xic. Thus, using (2.4), we obtain

fbm,L(ηL+ sgn(δn)xi)∆x− Z x_i+1

xi

fbm,L(ηL+ sgn(δn)x)dx

≤ε∆x=εq|δn|,

for alli∈Jn andn≥n2. Also, sincefbm,Lis bounded, there is a constantM such that

fb_m,L(ηL+ sgn(δ_n)x_i)∆x− Z xi+1

x_i

fb_m,L(ηL+ sgn(δ_n)x)dx

≤M∆x=M q|δ_n|,

for alli /∈J_n andn≥n₂. Therefore,

αn−1

X

i=0

fbm,L(ηL+ sgn(δn)xi)∆x= Z x_αn

x₀

fbm,L(ηL+ sgn(δn)x)dx+Rn,

where

|Rn| ≤(ε|Jn|+M(α_n− |Jn|))q|δn|.

Note thatα_n− |J_n|is the number of times that the monotonic sequence, {ηL+ sgn(δ_n)x_i}^α_i=0ⁿ,

crosses an integer. Thus, αn − |Jn| ≤ |xα_n −x0|+ 1 = αnq|δn|+ 1. Combined with

|Jn| ≤α_nandα_n≤a_nt/q, we have

|Rn| ≤εan|δn|t+M qan|δn|²t+M q|δn|.

Hence, sincean→ ∞, we have

lim sup

n→∞

|Rn| a_n|δn| ≤εt.

Sinceεwas arbitrary, it follows that

n→∞lim β(m, n) =e 1 q

q

X

η=1

n→∞lim 1 an|δn|

Z x_αn

x₀

fb_m,L(ηL+ sgn(δ_n)x)dx

.

Now, note thatx₀=η|δn|and

xα_n= (αnq+η)|δn|= (bantc −rn+η)|δn|.

Since|η−r_n| ≤q, we have|xαn−a_n|δn|t| ≤(q+ 1)|δn|. Thus, sincea_n→ ∞andfb_m,Lis bounded, we have

n→∞lim β(m, n) =e 1 q

q

X

η=1

n→∞lim 1 an|δn|

Z a_n|δn|t

0

fbm,L(ηL+ sgn(δn)x)dx

. (2.5)

Step 2. Assume L ∈ Q^and an|δn| → ∞. Then, taking into account that the function fb_m,L has period one, we can write

Z a_n|δn|t

0

fbm,L(ηL+ sgn(δn)x)dx

=ban|δn|tc Z 1

0

fbm,L(x)dx+

Z a_n|δn|t−ban|δn|tc

0

fbm,L(ηL+ sgn(δn)x)dx.

From (2.5) and the fact thatfbm,Lis bounded, we then obtain

n→∞lim β(m, n) =e 1 q

q

X

η=1 n→∞lim

ban|δn|tc an|δn|

Z 1

0

fbm,L(x)dx

+ 1

an|δn|

Z an|δn|t−ban|δn|tc

0

fb_m,L(ηL+ sgn(δ_n)x)dx

=t Z 1

0

fb_m,L(x)dx.

By (2.3) and the fact thatfL=P

m∈Zfm,Lis periodic with period 1, this gives

n→∞lim Sn(t) = 3t 4L

Z 1

0

fL(x)dx.

In light of (2.2), this completes half the proof of Theorem 2.2(ii). To complete the proof of Theorem 2.2(ii), it remains only to consider the caseL /∈Q, and this will be done in the final step of this proof.

Step 3.AssumeL∈Q^,an|δn| →k∈(0,∞), andδn>0for alln. From (2.5), we have

n→∞lim β(m, n) =e 1 q

q

X

η=1

1 k

Z kt

0

fbm,L(ηL+x)dx

.

From (2.3), the fact thatf_L has period 1, the identity L = p/q, and the substitution x=ks, this gives

n→∞lim Sn(t) = 3 4Lk

Z kt

0

1 q

q

X

η=1

fL(ηL+x)dx

= Z t

0

3 4p

q

X

η=1

f_L(ηL+ks)ds.

Step 4.AssumeL∈Q^,an|δn| →k∈(0,∞), andδn<0for alln. As in Step 3, we have

n→∞lim S_n(t) = Z t

0

3 4p

q

X

η=1

f_L(ηL−ks)ds.

By Lemma 2.1,

n→∞lim S_n(t) = Z t

0

3 4p

q

X

η=1

f_L((q−η+ 1)L+ks)ds

= Z t

0

3 4p

q

X

η=1

fL(ηL+ks)ds.

Step 5. We now prove Theorem 2.2(i). From the discussion following the statement of Theorem 2.2(i), we have that Theorem 2.2(i) with k = 0is equivalent to Theorem 1.1(i). Thus, we may assumean|δn| →k ∈ (0,∞). Let {Sn_m}_m∈N be any subsequence of {Sn}_n∈N. Recall from Step 1 that δn 6= 0 for all n ≥ n1. Choose a subsequence {Sn_m(j)}j∈Nof{Sn_m}m∈Nsuch thatsgn(δn_m(j))does note depend onj. By Steps 3 and 4,

j→∞lim Sn_m(j)(t) = Z t

0

3 4p

q

X

η=1

fL(ηL+ks)ds.

Since every subsequence has a subsequence converging to this limit, there is only one accumulation point, and sinceRis complete, it follows that

n→∞lim S_n(t) = Z t

0

3 4p

q

X

η=1

f_L(ηL+ks)ds.

Note thatηL=ηp/qand, sincepandqare relatively prime, the set of fractional parts {{ηp/q}: 1≤η≤q}, coincides with the set{j/q: 1≤j≤q}. Thus,

n→∞lim S_n(t) = Z t

0

3 4p

q

X

j=1

f_L j

q +ks

ds.

By (2.2), this completes the proof of Theorem 2.2(i).

Step 6. We now prove Theorem 2.2(ii). From Step 2, it suffices to considerL /∈Q^{. As} in the proof of the equidistribution theorem, the idea is to approximate the1-periodic functionf_m,Lby its truncated Fourier series.

Fixm∈Z^{and let}ε >0be arbitrary. Set FN(x) =

N

X

k=−N

cke^2πikx,

where

c_k= Z 1

0

f_m,L(y)e^2πikydy.

Sincekfm,Lk∞≤8(see [1, (2.23)]), we have|ck| ≤8.

The functionf_m,L is Hölder continuous of order1/3. Therefore, by Jackson’s theo- rem, the sequenceF_N converges uniformly on[0,1]tof_m,L, and we may chooseN ∈N such that for allx∈[0,1],

|FN(x)−fm,L(x)|< ε.

Recalling that{x}=x− bxc, we then have for any fixedt >0,

β(m, n) =e 1 an

bantc

X

j=1

f_m,L({jLn}) = 1 an

bantc

X

j=1

F_N({jLn}) +O(ε)

= 1 an

N

X

k=−N

ck ba_ntc

X

j=1

e^2πikjLn+O(ε).

In the above and for the remainder of this proof, the coefficients implied by the big O notation depend only ont.

Note that for any integerM ≥1and for any complex numberα,

M

X

j=1

α^j=

(_α(1−αM)

1−α ifα6= 1,

M ifα= 1. ^(2.6)

Setσk,n=Pbantc

j=1 e^2πikjLn. Then,σk,n=bantcifkLn∈Z^{. If}kLn∈/Z^{, then}

|σk,n|=

e^2πikLn1−e^2πikLnbantc 1−e^2πikLn

≤ 2

|1−e^2πikLn|.

SinceLn converges toL, which is irrational, we havelim_n→∞|1−e^2πikLn|>0. There- fore, there exists δ(k) > 0 and n0(k) ∈ N such that for all n ≥ n0(k), we have|1− e^2πikLn| ≥ δ(k). Note that for any suchn, we must havekL_n ∈/ Z^{, since} kL_n ∈ Z^im- plies|1−e^2πikLn|= 0. Now letδ= min_−N≤k≤Nδ(k)andn₀= max_−N≤k≤Nn₀(k). Then

|σk,n| ≤2/δwhenevern≥n0andk∈ {−N, . . . , N}.

Recall that Ln = pn/qn, where pn and qn are relatively prime numbers. Hence, kLn ∈Zif and only ifqn |k. Therefore, we obtain

β(m, n) =e 1 an

N

X

k=−N

ckσk,n+O(ε)

= 1 an

N

X

k=−N qn|k

ckbantc+Rn+O(ε),

where

|Rn|=

1 an

N

X

k=−N qn-k

ckσk,n

≤ 2 anδ

N

X

k=−N

|ck|=O(N a⁻¹_n δ⁻¹).

By (2.6),

1 qn

qn

X

j=1

(e^2πik/qn)^j=

(1 ifq_n|k, 0 ifqn-k.

As a consequence, we can write

β(m, n) =e ba_ntc an

N

X

k=−N

c_k 1 qn

q_n

X

j=1

e^2πikj/qn+O(N a⁻¹_n δ⁻¹) +O(ε)

=bantc anqn

qn

X

j=1

FN(j/qn) +O(N a⁻¹_n δ⁻¹) +O(ε)

=ba_ntc anqn

q_n

X

j=1

f_m,L(j/q_n) +O(N a⁻¹_n δ⁻¹) +O(ε).

In [1], it is shown thatqn → ∞whenL /∈ Q. Thus, lettingntend to infinity and using the fact thatfm,L is Riemann integrable on[0,1]gives

lim sup

n→∞

β(m, n)e −t Z 1

0

f_m,L(x)dx

=O(ε).

Sinceεwas arbitrary, and from (2.3) and (2.2), this completes the proof.

Examples. Here are some examples that were not covered by the results of [1]. Sup- pose thata_n = n² and b_n = (n+ 1)². In this caseL_n → 1 and a_n|δ_n| = |b_n−a_nL| = 2n+ 1→ ∞. Therefore,

ρ(t) =3 4

Z 1

0

f1(x)dx.

Ifan= 2nandbn= 3n+1, thenLn→3/2andan|δn|=|bn−anL|= 1for alln. Therefore, ρ(t) =1

4

f_3/2 1

2 +t

+f_3/2(t)

.

References

[1] Krzysztof Burdzy, David Nualart, and Jason Swanson,Joint convergence along different sub- sequences of the signed cubic variation of fractional Brownian motion, Probability Theory and Related Fields (2013), 1–36 (English).

[2] Krzysztof Burdzy and Jason Swanson,A change of variable formula with Itô correction term, Ann. Probab.38(2010), no. 5, 1817–1869. MR-2722787

[3] I. Nourdin and G. Peccati, Normal approximations with malliavin calculus: From stein’s method to universality, Cambridge University Press, 2012.

[4] Ivan Nourdin and Anthony Réveillac,Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical caseH = 1/4, Ann. Probab. 37(2009), no. 6, 2200–2230. MR-2573556

[5] Ivan Nourdin, Anthony Réveillac, and Jason Swanson,The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6, Electron. J. Probab. 15 (2010), no. 70, 2117–2162. MR-2745728

[6] D. Nualart and S. Ortiz-Latorre,Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stochastic Process. Appl.118(2008), no. 4, 614–628. MR-2394845 [7] Giovanni Peccati and Ciprian A. Tudor,Gaussian limits for vector-valued multiple stochastic

integrals, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., vol. 1857, Springer, Berlin, 2005, pp. 247–262. MR-2126978

[8] Jason Swanson, The calculus of differentials for the weak stratonovich integral, Malliavin Calculus and Stochastic Analysis (Frederi Viens, Jin Feng, Yaozhong Hu, and Eulalia Nualart˘a, eds.), Springer Proceedings in Mathematics & Statistics, vol. 34, Springer US, 2013, pp. 95–

111 (English).

[9] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York, 1982. MR-648108

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