# A shrinking target theorem for ergodic transformations of the unit interval

<EMAIL_ADDRESS>
###### Abstract.

We show that for any ergodic Lebesgue measure preserving transformation
$f:[0,1)\rightarrow[0,1)$ and any decreasing sequence
$\\{b_{i}\\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the
set

$\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b_{i}))$

has full Lebesgue measure for almost every $x\in[0,1)$ and almost every
$\alpha\in[0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at
$x\in[0,1)$ and $R_{\alpha}:[0,1)\rightarrow[0,1)$ is rotation by
$\alpha\in[0,1)$. As a corollary, we provide partial answer to a question
asked by Chaika (Question $3$, [1]) in the context of interval exchange
transformations.

###### Key words and phrases:

Shrinking target problem, Interval exchange transformations

###### 2020 Mathematics Subject Classification:

37A05, 37A10, 37E10

Shrey Sanadhya

Department of Mathematics

Ben Gurion University of the Negev

Be’er Sheva, 8410501, Israel.

##  1\.  Introduction

In this paper, we study a shrinking target problem in the context of ergodic
Lebesgue measure preserving transformations of the unit interval. In
particular, we consider the case when the shrinking targets are centered along
the orbit of an irrational rotation.

A typical shrinking target problem involves a probability measure preserving
dynamical system $(X,\mu,T)$ and a sequence of measurable sets
$\\{B_{n}\\}_{n\in{\mathbb{N}}}$ such that $\mu(B_{n})\rightarrow 0$ (which
are called shrinking targets). We ask if

$\mu(\\{x\in X:T^{n}x\in B_{n}\,\,\textrm{for infinitely
many}\,\,n\in\mathbb{N}\\})=1$

for $\mu$-a.e. $x\in X$. The Borel-Cantelli lemma gives us the necessary
condition i.e. $\underset{n=1}{\overset{\infty}{\sum}}\mu(B_{n})=\infty$.
Hence in the setting of a metric space, shrinking balls with fixed centers and
divergent radii sum, form a natural starting point for such a study. One of
the earliest result in this setting is due to J. Kurzweil [2], who studied the
shrinking target problem in the context of irrational rotations.

For $\alpha\in[0,1)$, we define $R_{\alpha}\colon[0,1)\to[0,1)$ as
$R_{\alpha}(x)=x+\alpha-\lfloor x+\alpha\rfloor$. Let $B(x,r)$ be the ball of
radius $r$ centered at $x\in[0,1)$ and $\lambda$ be the Lebesgue measure on
$[0,1)$. Kurzweil proved the following result :

######  Theorem 1.1 (J. Kurzweil [2]).

For any decreasing sequence of positive real numbers
$\\{b_{i}\\}_{i=1}^{\infty}$ with divergent sum and for almost every
$\alpha\in[0,1)$,

$\lambda\left(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,B(R_{\alpha}^{i}(x),b_{i})\right)=1$

every $x\in[0,1)$.

The pioneering work of Kurzweil motivated the study of shrinking target
problem in diverse settings. We refer interested readers to
[3],[4],[5],[6],[7] and the survey paper [8].

J. Chaika [1] studied shrinking target problem in context of interval exchange
transformations (IETs) (see Definition 2.1). He showed that as a family,
interval exchange transformations satisfy the shrinking target property (see
Theorem 2.2). This paper is motivated by results in [1]. In particular, we
provide a partial answer to Question $3$ of [1]. We mention the question below
for completion.

######  Question 1.2.

Let $\\{y_{i}\\}_{i=1}^{\infty}$ be a sequence of points in $[0,1)$ and
$\\{b_{i}\\}_{i=1}^{\infty}$ be a sequence of positive real numbers with
divergent sum. Is it true that for almost every IET $T$, we have
$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,T^{-i}(B(y_{i},b_{i})))=1$?

We answer the above question for the case when the sequence $\\{y_{i}\\}$ is
an orbit under irrational rotations. To our surprise, if the shrinking targets
are centered at points along the orbit of an irrational rotation, Question 1.2
can be answered in much more generality, i.e. for any ergodic Lebesgue measure
preserving transformation of the unit interval and any sequence
$\\{b_{i}\\}_{i=1}^{\infty}$ of positive real numbers with divergent sum (see
Theorem 2.5). Thus, we obtain a partial answer to Chaika’s question as a
corollary (see Corollary 2.6).

The outline of the paper is as follows. In Section 2, we provide background
and state the main results (Theorem 2.5 and Corollary 2.6). Section 3 is
dedicated to the proof of Theorem 2.5.

##  2\. Preliminaries and statement of results

######  Definition 2.1.

Let $P=(p_{1},p_{2},...,p_{d})$, where $p_{i}\geq 0$ for each
$i\in\\{1,2,...,d\\}$ be a $d$-dimensional vector which defines interval
$I=[0,\underset{i=1}{\overset{d}{\sum}}p_{i})$, with $d$ sub intervals

$I_{1}=[0,p_{1}),I_{2}=[p_{1},p_{1}+p_{2}),....,I_{d}=[p_{1}+...+p_{d-1},p_{1}+...+p_{d-1}+p_{d}).$

Let $\pi$ be a fixed permutation on the set $\\{1,2,...,d\\}$. An interval
exchange transformation (IET) is a map $T:I\rightarrow I$ which permutes the
$d$ sub intervals $I_{i}$ by $\pi$. In other words for $x\in I_{j}$

$T(x)=x-\underset{t<j}{\sum}p_{t}+\underset{\pi(t^{\prime})<\pi(j)}{\sum}p_{t^{\prime}}.$

The IET defined above is on $d$-intervals and will be referred to as a
$d$-IET. In this paper we will restrict ourselves to IETs on unit the
interval, i.e. $I=[0,1)$ for convenience. A permutation $\pi\in S_{d}$ is
called irreducible if $\pi(\\{1,...,t\\})\neq\\{1,...,t\\}$ for any $t<d$. It
was shown by M. Keane [9] that IETs with dense orbits have irreducible
permutations. Hence these IETs are important from the point of view of
shrinking target property. In this paper we will work with $d$-IETs (for
$d>2$) with a fixed irreducible permutation $\pi\in S_{d}$ on the unit
interval $I=[0,1)$. We can parametrize such IETs by the $d$-dimensional
simplex $\Delta^{d}=\\{(p_{1},p_{2},...,p_{d}):p_{i}\geq
0,\underset{i}{\sum}\,p_{i}=1\\}$.

Note that the standard simplex $\Delta^{d}$ comes equipped with Lebesgue
measure. In this paper for a fixed irreducible permutation, the term almost
every IET will refer to the Lebesgue measure on simplex $\Delta^{d}$. We will
consider it as the parameterizing space of IETs in this paper. S. Kerkchoff,
H. Masur, and J. Smillie (see [10]) showed that for almost every IET is
uniquely ergodic with respect to $\lambda$ (Lebesgue measure on the interval
$[0,1)$). Below we list some known shrinking target results for IETs.

######  Theorem 2.2 (J. Chaika [1]).

For almost every IET $T$ and a decreasing sequence
$\\{b_{i}\\}_{i=1}^{\infty}$ of real numbers with divergent sum,

$\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,T^{-i}\left(B(y,b_{i})\right)$
has full $\lambda$-measure for every $y$.

Following logarithm law is due to S. Galatolo [11].

######  Theorem 2.3 (S. Galatolo [11] ).

Given an IET $T$ let

${\tau_{r}(x,y)=\min\\{n\in\mathbb{N},n>0:|T^{n}x-y|<r\\}}.$

For almost every IET $T$, ${\underset{r\to
0}{\mathrm{lim\,inf}}\,\,\frac{\log(\tau_{r}(x,y))}{-\log r}=1}$ for almost
every $x$.

Following result is due to L. Marchese [12].

######  Theorem 2.4 (L. Marchese [12]).

Let $\\{b_{i}\\}_{i=1}^{\infty}$ be a decreasing sequence with divergent sum
and with the additional property that $\\{ib_{i}\\}_{i=1}^{\infty}$ is
decreasing. For almost every IET $T$

$\delta\in\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,B(T^{i}(\delta^{\prime}),b_{i}))$

where $\delta$ and $\delta^{\prime}$ are any discontinuities of $T$.

Our main result is the following :

######  Theorem 2.5.

Let $f:[0,1)\rightarrow[0,1)$ be an ergodic measure preserving transformation
(with respect to $\lambda$). Then for a decreasing sequence
$\\{b_{i}\\}_{i=1}^{\infty}$ of real numbers with divergent sum

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b_{i})))=1$

for almost every $x\in[0,1)$ and almost every $\alpha\in[0,1)$.

Corollary 2.6 follows directly from Theorem 2.5.

######  Corollary 2.6.

For every ergodic IET $T$ and any decreasing sequence
$\\{b_{i}\\}_{i=1}^{\infty}$ of real numbers with divergent sum

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,T^{-i}(B(R^{i}_{\alpha}x,b_{i})))=1$

for almost every $x\in[0,1)$ and almost every $\alpha\in[0,1)$.

##  3\. Proof of Theorem 2.5

Let $f:[0,1)\rightarrow[0,1)$ be an ergodic measure preserving transformation
(with respect to $\lambda$). For $n\in\mathbb{N},\,m\in\mathbb{Z}$, we define
sets $U_{n,m}$, $V_{n}$ and $W$ as follows:

$U_{n,m}=\\{(\alpha,y)\in[0,1)\times\mathbb{R}:|f^{n}y-n\alpha-m|\leq
b_{n}\\};\,\,n\in\mathbb{N},\,m\in\mathbb{Z}.$ (3.1)

######  Remark 3.1.

For the definition of $U_{n,m}$ to make sense we need to extend the domain of
$f^{n}$ to ${\mathbb{R}}$. We do it as follows : Let $y\in{\mathbb{R}}$, and
$\lfloor y\rfloor=y\,(\mathrm{mod}\,1)$, then $y=\lfloor y\rfloor\,+\,k$ for
some $k\in{\mathbb{Z}}$. We define $f^{n}(y)=f^{n}(\lfloor y\rfloor)+k$. With
this new definition $f^{n}$ is well defined over entire $\mathbb{R}$.

$V_{n}=\underset{m=-\infty}{\overset{\infty}{\bigcup}}U_{n,m}.$ (3.2)

$W=\underset{s=1}{\overset{\infty}{\bigcap}}\underset{n=s}{\overset{\infty}{\bigcup}}V_{n}.$
(3.3)

######  Proposition 3.2.

Let $\\{b_{n}\\}_{n=1}^{\infty}$ be a decreasing sequence of positive real
numbers with divergent sum, then there exists a decreasing sequence
$\\{b^{\prime}_{n}\\}_{n=1}^{\infty}$ (of positive real numbers with divergent
sum) such that
$\underset{n\rightarrow\infty}{\mathrm{lim}}\,\dfrac{b^{\prime}_{n}}{b_{n}}=0$
and $b^{\prime}_{n}<\dfrac{1}{16\,n}$ for every $n\in\mathbb{N}$.

Proof. We construct a sequence $\\{c_{n}\\}_{n=1}^{\infty}$ as follows : Since
$\underset{n=1}{\overset{\infty}{\sum}}b_{n}=\infty$, there exists
$n_{2}\in\mathbb{N}$ such that $\underset{n=1}{\overset{n_{2}}{\sum}}b_{n}\geq
2$, define $c_{n}=\dfrac{b_{n}}{2}$ for $1\leq n\leq n_{2}$. Similarly, there
exists $n_{3}\in\mathbb{N}$ such that
$\underset{n=n_{2}+1}{\overset{n_{3}}{\sum}}b_{n}\geq 3$, define
$c_{n}=\dfrac{b_{n}}{3}$ for $n_{2}+1\leq n\leq n_{3}$ and so on. Put
$b^{\prime}_{n}=\mathrm{min}\,\,\\{\dfrac{1}{16\,n}\,,\,c_{n}\\}$ for
$n\in\mathbb{N}$, then the sequence $\\{b^{\prime}_{n}\\}_{n=1}^{\infty}$
satisfies the required conditions. $\square$

By replacing $\\{b_{n}\\}$ with $\\{b^{\prime}_{n}\\}$ we define
$U^{\prime}_{n,m}$, $V^{\prime}_{n}$ and $W^{\prime}$ in the same manner as
above. Let $S$ be the unit square $S=\\{(\alpha,y):0\leq\alpha<1,0\leq y<1\\}$
and $K$ be the strip $K=\\{(\alpha,y):0\leq\alpha<1,y\in\mathbb{R}\\}$. We
denote by $\lambda_{2}$ the Lebesgue measure on $(0,1]^{2}$.

######  Proposition 3.3.

For every $n\in\mathbb{N}$

$\lambda_{2}(S\cap V^{\prime}_{n})=\lambda_{2}(K\cap
U^{\prime}_{n,0})=2b^{\prime}_{n}.$

Proof. Observe that the set $V^{\prime}_{n}$ consists of countable number of
copies of set $U^{\prime}_{n,0}$, shifted by integer values in the $y$
direction. Hence the area $K\cap U^{\prime}_{n,0}$, (where $K$ is the vertical
strip in $\alpha\times y$ plane as defined above) is equal to the area $S\cap
V^{\prime}_{n}$, where $S$ is the unit square in the $\alpha\times y$ plane).

For $n\in\mathbb{N}$, $m\in\mathbb{Z}$, we define following sets:

$P^{\prime}_{n,m}=\\{(\alpha,f^{n}y)\in[0,1)\times\mathbb{R}:|f^{n}y-n\alpha-m|\leq
b^{\prime}_{n}\\};\,\,n\in\mathbb{N},\,m\in\mathbb{Z}.$ (3.4)

$Q^{\prime}_{n}=\underset{m=-\infty}{\overset{\infty}{\bigcup}}P^{\prime}_{n,m}.$
(3.5)

Denote by $z_{y}=f^{n}y$, for all $y\in\mathbb{R}$. Thus we can write

$P^{\prime}_{n,m}=\\{(\alpha,z_{y})\in[0,1)\times\mathbb{R}:|z_{y}-n\alpha-m|<b^{\prime}_{n}\\};\,\,\,Q^{\prime}_{n}=\underset{m=-\infty}{\overset{\infty}{\bigcup}}P^{\prime}_{n,m}.$
(3.6)

We will show

$\lambda_{2}(S_{z}\cap Q^{\prime}_{n})=\lambda_{2}(K_{z}\cap
P^{\prime}_{n,0})=2b^{\prime}_{n},$ (3.7)

where $S_{z}$ is the square $S_{z}=\\{(\alpha,z_{y}):0\leq\alpha<1,0\leq
z_{y}<1\\}$ and $K_{z}$ is the strip
$K_{z}=\\{(\alpha,z_{y}):0\leq\alpha<1,z_{y}\in\mathbb{R}\\}$ on the
$\alpha\times z$ plane. Note that
$P^{\prime}_{n,0}=\\{(\alpha,z_{y}):|z_{y}-n\alpha|<b^{\prime}_{n}\\}$. Thus
$\lambda_{2}(K_{z}\cap P^{\prime}_{n,0})$ is the area of the region
$|z_{y}-n\alpha|<b^{\prime}_{n}$, $\alpha\in[0,1)$. This area is given by

$\int_{0}^{1}(b^{\prime}_{n}+n\alpha)-(n\alpha-b^{\prime}_{n})\,d\alpha=2b^{\prime}_{n}$

Since $f^{n}$ is measure preserving $\lambda_{2}(S\cap
V^{\prime}_{n})=\lambda_{2}(S_{z}\cap Q^{\prime}_{n})$. $\square$

######  Proposition 3.4.

For $j,k\in\mathbb{N}$ such that $j<k$ we have

$\lambda_{2}(S\cap V^{\prime}_{j}\cap
V^{\prime}_{k})=4b^{\prime}_{j}b^{\prime}_{k}.$

Proof. Observe that for $j<k$,

$\lambda_{2}(S_{z}\cap Q^{\prime}_{j}\cap
Q^{\prime}_{k})=\lambda_{2}((K_{z}\cap P^{\prime}_{k,0})\cap Q^{\prime}_{j}).$
(3.8)

Since $f^{n}$ is measure preserving $(\ref{eq ints})$ is equivalent to

$\lambda_{2}(S\cap V^{\prime}_{j}\cap V^{\prime}_{k})=\lambda_{2}((K\cap
U^{\prime}_{k,0})\cap V^{\prime}_{j}).$ (3.9)

Again put $z_{y}=f^{n}y$, for all $y\in\mathbb{R}$ and consider the definition
of $P^{\prime}_{n,m}$ as given in $(\ref{eq defP})$. Observe that set
$(K_{z}\cap P^{\prime}_{k,0})\cap Q^{\prime}_{j}$ consists of $k-j$
parallelograms each of area equal to
$\dfrac{4b^{\prime}_{j}b^{\prime}_{k}}{(k-j)}$. Thus

$\lambda_{2}((K_{z}\cap P^{\prime}_{k,0})\cap
Q^{\prime}_{j})=4b^{\prime}_{j}b^{\prime}_{k}.$ (3.10)

The equivalence of $(\ref{eq ints})$ and $(\ref{eq ints2})$ completes the
proof. $\square$

######  Lemma 3.5.

Let $t\in\mathbb{N}$ be a fixed positive integer. Then for every
$N_{0}\in\mathbb{N}$, there exists a positive integer $N$ such that

$\lambda_{2}\Big{(}S\cap\Big{(}\underset{n=N_{0}}{\overset{N}{\bigcup}\,V^{\prime}_{tn}}\Big{)}\Big{)}>\frac{1}{8}.$

Proof. Note the following inequality:

$\lambda_{2}\Big{(}S\cap\Big{(}\underset{n=N_{0}}{\overset{N}{\bigcup}\,V^{\prime}_{tn}}\Big{)}\Big{)}\geq\underset{n=N_{0}}{\overset{N}{\sum}}\lambda_{2}(S\cap
V^{\prime}_{tn})\,\,-\underset{N_{0}\leq j<k\leq N}{\sum}\lambda_{2}(S\cap
V^{\prime}_{tj}\cap V^{\prime}_{tk}).$ (3.11)

By Proposition 3.3 and Proposition 3.4 for every $t\in\mathbb{N}$,

$\lambda_{2}(S\cap
V^{\prime}_{tn})=2b^{\prime}_{tn}\,;\,\,\,\mathrm{and}\,\,\lambda_{2}(S\cap
V^{\prime}_{tj}\cap V^{\prime}_{tk})=4\,\,b^{\prime}_{tj}\,\,b^{\prime}_{tk}.$

Which implies

$\lambda_{2}\Big{(}S\cap\Big{(}\underset{n=N_{0}}{\overset{N}{\bigcup}\,V^{\prime}_{tn}}\Big{)}\Big{)}\geq
2\underset{n=N_{0}}{\overset{N}{\sum}}b^{\prime}_{tn}-\,\,4\underset{N_{0}\leq
j<k\leq N}{\sum}b^{\prime}_{tj}\,\,b^{\prime}_{tk}\geq
2\underset{n=N_{0}}{\overset{N}{\sum}}b^{\prime}_{tn}-\Big{(}2\underset{n=N_{0}}{\overset{N}{\sum}}b^{\prime}_{tn}\Big{)}^{2}.$

Note that that for a fixed $t\in{\mathbb{Z}}$,
$\underset{n=0}{\overset{\infty}{\sum}}b^{\prime}_{tn}=\infty$. To see this,
assume by contradiction that there exists $c\in\mathbb{R}$, such that
$\underset{n=0}{\overset{\infty}{\sum}}b^{\prime}_{tn}\leq c$. Using the fact
that $\\{b^{\prime}_{n}\\}$ is a decreasing sequence we get,

$\underset{n=t}{\overset{\infty}{\sum}}b^{\prime}_{n}\leq
tb^{\prime}_{t}+tb^{\prime}_{2t}+tb^{\prime}_{3t}+...+=t\,\underset{n=1}{\overset{\infty}{\sum}}b^{\prime}_{tn}\leq
t\,c$

which is a contradiction since $\\{b^{\prime}_{n}\\}$ has divergent sum.

By definition for $n\in\mathbb{N}$, $b^{\prime}_{n}<\dfrac{1}{16\,n}$. Thus,
$b^{\prime}_{tn}<\dfrac{1}{16\,tn}\leq\dfrac{1}{16}\,\,\textrm{for}\,\,t,n\in\mathbb{N}.$
Hence we can choose the index $N$ such that

$\frac{3}{8}<2\underset{n=N_{0}}{\overset{N}{\sum}}b^{\prime}_{tn}<\frac{3}{8}+2.\frac{1}{16}=\frac{1}{2}.$

Thus we have

$\lambda_{2}\Big{(}S\cap\Big{(}\underset{n=N_{0}}{\overset{N}{\bigcup}\,V^{\prime}_{tn}}\Big{)}\Big{)}>\frac{3}{8}-(\frac{1}{2})^{2}>\frac{3}{8}-\frac{1}{4}=\frac{1}{8}.$

$\square$

######  Proposition 3.6.

For almost all $\alpha\in[0,1)$ and a positive measure set of $x\in[0,1)$

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b^{\prime}_{i})))>0.$
(3.12)

Before we prove Proposition 3.6 we discuss the following lemma:

######  Lemma 3.7.

Let $B\subset[0,1)$ be a set such that $\lambda(B)>\sigma$ for some
$0<\sigma<1$. Assume that there exists $k\in\mathbb{N}$ such that

$B=B+\dfrac{1}{k}\,\,(\mathrm{mod}\,\,1)$

Let $J\subset[0,1)$ be an interval such that
$\dfrac{100}{k}<l(J)<\dfrac{101}{k}$, where $l(J)$ denotes the length of $J$.
Then there exists a constant $C$ (independent of $J$ and $k$) such that

$\lambda(J\cap B)>C\,.\,\sigma\,.\,l(J).$

Proof. Let $x\in B$ then there exists $n(x)\in\mathbb{N}$, such that
$y_{x}:=x+\dfrac{n(x)}{k}\,\,(\mathrm{mod}\,\,1)\in J$. This is true since
$l(J)$ is much larger than $\dfrac{1}{k}$. Since $l(J)>\dfrac{100}{k}$,
$y_{x}$ is not unique, in fact there are at least $99$ such points (taking
into consideration end points of $J$). Define

$D_{x}=\\{y_{x}\in J:y_{x}=x+\dfrac{n}{k},\,\textrm{for
some}\,\,n\in\mathbb{N}\\}.$

As discussed above $|D_{x}|$ is at least $99$. Since $B$ is invariant under
$\dfrac{1}{k}$ we have $D_{x}\subset B$. Observe that

$B=\underset{j=0}{\overset{k-1}{\bigsqcup}}\big{(}\,B\cap[0,\frac{1}{k})\,\big{)}+\frac{j}{k}$

Thus we get

$\lambda(B\cap J)=\int_{B\cap\,[0,\frac{1}{k})}|D_{x}|\,\,dx$

$\geq 99.\,\,\,\lambda\big{(}B\cap[0,\frac{1}{k})\big{)}\geq
99.\,\,\dfrac{\sigma}{k}=\dfrac{99}{101}\,\,\sigma\,\,\dfrac{101}{k}>\dfrac{99}{101}\,\,\sigma\,\,l(J).$

Put $C=\dfrac{99}{101}$, we get

$\lambda(B\cap J)>C\,.\,\sigma\,.\,l(J).$

$\square$

Proof of Proposition 3.6. Note that by definition for $t\in\mathbb{N}$,

$U^{\prime}_{tn,m}=\\{(\alpha,y)\in[0,1)\times\mathbb{R}:|f^{tn}y-tn\alpha-m|\leq
b^{\prime}_{tn}\\};\,\,n\in\mathbb{N},\,m\in\mathbb{Z}.$ (3.13)

$V^{\prime}_{tn}=\underset{m=-\infty}{\overset{\infty}{\bigcup}}U^{\prime}_{tn,m}.$
(3.14)

For $t\in\mathbb{N}$ we define

$W^{\prime}_{t}=\underset{s=1}{\overset{\infty}{\bigcap}}\underset{n=s}{\overset{\infty}{\bigcup}}V^{\prime}_{tn}.$
(3.15)

By Lemma 3.5 we have $\lambda_{2}(S\cap W^{\prime}_{t})>0$. If we think of
$U^{\prime}_{tn,m}$ as

$U^{\prime}_{tn,m}=\\{(\alpha,y)\in[0,1)\times\mathbb{R}:|f^{tn}y-tn\alpha-m|\leq
b^{\prime}_{tn}\\};\,\,n\in\mathbb{N},\,m\in\mathbb{Z}.$

Then $\lambda_{2}(S\cap W^{\prime}_{t})>0$ implies that for every
$t\in\mathbb{N}$, for a positive measure set of $\alpha\in[0,1)$ and a
positive measure set of $x\in[0,1)$

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-ti}(B(R_{\alpha}^{ti}x,b^{\prime}_{ti})))>0.$
(3.16)

For $t\in\mathbb{N}$ we call such positive set of $\alpha\in[0,1)$ by $A_{t}$.
Thus in particular for $\alpha\in A_{1}$, we have

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b^{\prime}_{i})))>0.$
(3.17)

for a positive measure set of $x\in[0,1)$. Note that for every
$t\in\mathbb{N}$ the set $A_{t}$ is invariant under addition by
$\dfrac{1}{t}\,\,(\mathrm{mod}\,1)$; and $A_{t}\subseteq A_{1}$. Lemma 3.5
implies that for every $t\in\mathbb{N}$, $\lambda(A_{t})>\dfrac{1}{8}$. We
want to show that $A_{1}$ has full measure.

For any $2r\in(0,1)$, we can find appropriate $t_{r}\in\mathbb{N}$ such that
$\dfrac{100}{t_{r}}<2r<\dfrac{101}{t_{r}}$. For any $x\in(0,1]$ by using Lemma
3.7 we get

$\lambda(A_{t_{r}}\cap B(x,r))>C\,.\,\dfrac{1}{8}\,.\,2r$

where $B(x,r)$ denotes a ball of radius $r$ centered at $x$. Since
$A_{t_{r}}\subset A_{1}$ we get

$\lambda(A_{1}\cap B(x,r))>C\,.\,\dfrac{1}{8}\,.\,2r\,=\,\dfrac{C}{4}\,r.$
(3.18)

Assume $\lambda(A_{1})<1$ and let $x$ be a Lebesgue density point for
$A_{1}^{c}$ then

$\underset{r\rightarrow 0}{\mathrm{lim}}\,\,\dfrac{\lambda(A_{1}^{c}\cap
B(x,r))}{\lambda(B(x,r))}=1.$

Thus for every $\epsilon>0$ we can find $r>0$ such that

$\lambda(A_{1}^{c}\cap B(x,r))>(1-\epsilon)\,2r.$

Choose $\epsilon=\dfrac{C}{8}$ (for $C$ as above) we get

$\lambda(A_{1}^{c}\cap
B(x,r))>(1-\dfrac{C}{8})\,2r=2r-2r.\dfrac{C}{8}=2r-\dfrac{C}{4}\,r.$

Hence we get,

$\lambda(A_{1}\cap B(x,r))<\dfrac{C}{4}\,r.$

which is a contradiction thus $\lambda(A_{1})=1$. This completes the proof of
Proposition 3.6. $\square$

######  Lemma 3.8.

For almost all $\alpha\in[0,1)$ and a.e $x\in[0,1)$

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b^{\prime}_{i})))=1.$

Proof. So far we have shown that for almost every $\alpha\in[0,1)$, and a
positive measure set of $x\in[0,1)$

$\lambda(\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b^{\prime}_{i})))>0.$
(3.19)

Fix an irrational $\alpha\in[0,1)$ from the full measure set of $\alpha$’s
that satisfies $(\ref{eq pos})$ and define the set $H_{\alpha}$ as follows :

$H_{\alpha}=\\{(x,y)\in[0,1)^{2}:y\in\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b^{\prime}_{i}))\\}$

Note that $(\ref{eq pos})$ implies that $\lambda_{2}(H_{\alpha})>0$ where
$\lambda_{2}$ is the Lebesgue measure on $[0,1)^{2}$. We show that if
$(x,y)\in H_{\alpha}$ then $(R_{\alpha}x,f(y))\in H_{\alpha}$. To see this,
let $(x,y)\in H_{\alpha}$ then

$f^{i}y$ belongs to an infinite number of sets
$B(R_{\alpha}^{i}x,b^{\prime}_{i})$.

Since $b^{\prime}_{i}\leq b^{\prime}_{i-1}$

$f^{i}y$ belongs to an infinite number of sets
$B(R_{\alpha}^{i}x,b^{\prime}_{i-1})$.

Thus we get

$f^{i-1}f(y)$ belongs to an infinite number of sets
$B(R_{\alpha}^{i-1}(R_{\alpha}x),b^{\prime}_{i-1})$.

Hence

$f(y)$ belongs to an infinite number of sets
$f^{-(i-1)}B(R_{\alpha}^{i-1}(R_{\alpha}x),b^{\prime}_{i-1})$.

In other words

$f(y)$ belongs to an infinite number of sets
$f^{-i}B(R_{\alpha}^{i}(R_{\alpha}x),b^{\prime}_{i})$

which implies $(R_{\alpha}x,f(y))\in H_{\alpha}$.

Now we mention a condition for ergodicity of cartesian product of two ergodic
measure preserving transformations of probability space. We refer our readers
to [13, Theorem 12] for a detailed proof. In the theorem below $U_{T}$ (resp.
$U_{S}$) denotes the Koopman operator associated with the transformation $T$
(resp. $S$).

Theorem (Ergodicity of product). Let $T$ and $S$ be two probability measure
preserving transformations. If $T$ and $S$ are ergodic, then $T\times S$ is
ergodic if and only if $U_{S}$ and $U_{T}$ have no common eigenvalues other
than $1$.

Since $f$ is a probability measure preserving transformation $U_{f}$ has at
most countably many eigenvalues. Hence $U_{f}$ shares non-trivial eigenvalue
with $R_{\alpha}$ for at most countably many $\alpha$. Thus by the result
above $R_{\alpha}\times f$ is ergodic for almost every $\alpha\in[0,1)$. This
implies $\lambda_{2}(H_{\alpha})=1$ for almost every $\alpha\in[0,1)$. This
completes the proof of the lemma. $\square$

Proof of Theorem 2.5. Observe that $b^{\prime}_{i}<b_{i}$, for every
$i\in\mathbb{N}$, hence

$\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,(B(R_{\alpha}^{i}x,b^{\prime}_{i}))\subset\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,(B(R_{\alpha}^{i}x,b_{i})).$

Thus we get,

$\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b^{\prime}_{i}))\subset\underset{n=1}{\overset{\infty}{\cap}}\,{\underset{i=n}{\overset{\infty}{\cup}}}\,f^{-i}(B(R_{\alpha}^{i}x,b_{i})).$

Hence Lemma 3.8 implies Theorem 2.5. $\square$

Acknowledgments. I would like to thank Jon Chaika for suggesting this problem
and many helpful discussions. I would also like to thank the referee for
careful reading and valuable suggestions.

## References

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# Power-law cross-correlations estimation under heavy tails

Ladislav Kristoufek<EMAIL_ADDRESS>Institute of Information Theory and
Automation, The Czech Academy of Sciences, Pod Vodarenskou Vezi 4, 182 08,
Prague 8, Czech Republic
Institute of Economic Studies, Faculty of Social Sciences, Charles University,
Opletalova 26, 110 00, Prague 1, Czech Republic

###### Abstract

We examine the performance of six estimators of the power-law cross-
correlations – the detrended cross-correlation analysis, the detrending
moving-average cross-correlation analysis, the height cross-correlation
analysis, the averaged periodogram estimator, the cross-periodogram estimator
and the local cross-Whittle estimator – under heavy-tailed distributions. The
selection of estimators allows to separate these into the time and frequency
domain estimators. By varying the characteristic exponent of the
$\alpha$-stable distributions which controls the tails behavior, we report
several interesting findings. First, the frequency domain estimators are
practically unaffected by heavy tails bias-wise. Second, the time domain
estimators are upward biased for heavy tails but they have lower estimator
variance than the other group for short series. Third, specific estimators are
more appropriate depending on distributional properties and length of the
analyzed series. In addition, we provide a discussion of implications of these
results for empirical applications as well as theoretical explanations.

###### keywords:

power-law cross-correlations, heavy tails, Monte Carlo study

††journal: Communications in Nonlinear Science and Numerical Simulation

##  1 Introduction

Power-law cross-correlations have brought a new perspective into analysis of
bivariate time series with applications across a wide range of disciplines –
hydrology [1], (hydro)meteorology [2, 3], seismology and geophysics [4, 5],
economics and finance [6, 7, 8, 9, 10, 11, 12], biometrics [13], biology [14],
DNA sequences [15], neuroscience [16], electricity [17], traffic [18, 19, 20,
21], and others. Analyzing cross-correlations between a pair of time series
brings new insights into their dynamics and specifically, the power-law
behavior of these may suggest very special features compared to exponential
(vanishing) cross-correlations [22, 23, 24, 25, 26, 27]. Presence of power-law
cross-correlations is a very current topic in various scientific fields but,
unfortunately, empirical papers strongly outnumber theoretical ones discussing
their emergence and origin. Podobnik et al. [28] suggest that power-law cross-
correlated processes can occur as a mixture of correlated long-range
correlated processes. Sela & Hurvich [25] and Kristoufek [26, 27] add a
possibility of long-range correlated processes with cross-correlated error
terms as sources of long-range cross-correlations together with a discussion
of various properties of such processes. Investigation of power-law cross-
correlated processes has also led to an important novelty in analyzing cross-
correlations at specific scales [29, 30, 31, 32] as well as a new regression
framework for specific scales [33].

Formally, processes labeled as power-law (long-range, long-term) cross-
correlated can be defined in both time and frequency domain. In the time
domain, the long-range cross-correlated processes $\\{x_{t}\\}$ and
$\\{y_{t}\\}$ are characterized by a power-law (hyperbolically) decaying
cross-correlation function $\rho_{xy}(k)$ with a time lag $k$ so that
$\rho_{xy}(k)\propto k^{2H_{xy}-2}$ for $k\rightarrow+\infty$ [34]. In the
frequency domain, the processes have a divergent at origin cross-power
spectrum. Specifically, the cross-power spectrum $f_{xy}(\lambda)$ with
frequency $\lambda$ scales as $|f_{xy}(\lambda)|\propto\lambda^{1-2H_{xy}}$
for $\lambda\rightarrow 0+$ [25]. The bivariate Hurst exponent $H_{xy}$ is a
measure of long-range cross-correlations, specifically their decay. For
$H_{xy}=0.5$, the processes are not considered long-range cross-correlated
whereas for $H_{xy}>0.5$, these are referred to as the cross-persistent
processes which tend to move together. Time series with $H_{xy}<0.5$ form a
very specific type of processes which have been theoretically only sparsely
examined.

In the same way as for the univariate series, the correlations scaling and the
underlying distribution tail behavior are tightly interconnected [35, 36].
Investigation of the relationship between the two goes back to Mandelbrot &
Wallis [37] – and thus hydrology – when examining the emerge of the so-called
Hurst effect [38]. Even though the effect emerges both for long-range
dependent processes and for processes with heavy tails, it is the correlation
function, and specifically its shape, which is crucial. The confusion about
the real and the spurious sources of the Hurst effect spreads also into self-
similarity and phase transitions [39]. Such spurious effects can easily
transfer into bivariate, or in general multivariate, setting which forms a
motivational cornerstone of this text.

Presence of heavy tails in distributions, or in other words high occurrence of
extreme events, is well documented across disciplines [40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50]. However, the current stream of literature does not take a
possible effect on estimators of the bivariate Hurst exponent implied by such
heavy-tailed (fat-tailed) distributions into consideration. Here we fill this
gap by focusing on three frequency domain estimators – the average periodogram
estimator [25], the cross-periodogram estimator and the local cross-Whittle
estimator [51] – and three time domain estimators – the detrended cross-
correlation analysis [52], the detrending moving-average cross-correlation
analysis [7, 53] and the height cross-correlation analysis [54] – performance
of which we examine under such distributions. Specifically, we study the
effect of varying characteristic exponent of the $\alpha$-stable distributions
on bias, variance and mean squared error of the estimators for various time
series lengths.

##  2 Methods

In this section, we shortly describe six estimators of the bivariate Hurst
exponent. These are separated into two groups based on their domain of
operation – time and frequency. The simulations setting is then described in
detail.

###  2.1 Time domain estimators

The detrended cross-correlation analysis (DCCA, or DXA) [52] is the most
popular time domain method of the bivariate Hurst exponent estimation,
constructed as a bivariate generalization of the detrended fluctuation
analysis (DFA) [55, 56]. The method has led to various generalizations and
expansions [57, 58, 59]. Considering two long-range cross-correlated series
$\\{x_{t}\\}$ and $\\{y_{t}\\}$ with $t=1,\ldots,T$ and their respective
profiles $\\{X_{t}\\}$ and $\\{Y_{t}\\}$, the DCCA procedure is based on an
examination of the detrending covariances scaling with respect to scale $s$.
Specifically, the time series are divided into overlapping boxes of length $s$
and the linear time trend is estimated in each box yielding
$\widehat{X_{k,j}}$ and $\widehat{Y_{k,j}}$ for boxes $j\leq k\leq j+s-1$. The
detrended covariance $f_{DCCA}^{2}(s,j)$ is obtained for each box $j$ of
length $s$ and it is further averaged over all boxes of length $s$ to get
$F^{2}_{DCCA}(s)$ as an estimated covariance for scale $s$. For the power-law
cross-correlated processes, we have $F_{DCCA}^{2}(s)\propto s^{2H_{xy}}$.
There are various ways of treating overlapping and non-overlapping boxes for
scales $s$ as the method can become computationally highly demanding [35, 36,
60, 61, 62]. Due to this fact, we use non-overlapping boxes with a minimum
scale of 10, a maximum scale of $T/5$ and a step between $s$ equal to 10 in
the simulations.

The height cross-correlation analysis (HXA) [54] is a bivariate generalization
of the height-height correlation analysis (HHCA) [63, 64] and the generalized
Hurst exponent approach (GHE) [65, 66, 67]. The method is based on scaling of
the height-height covariance function
$K_{xy,2}(\tau)=\frac{\nu}{T^{\ast}}\sum_{t=1}^{T^{\ast}/\nu}|\Delta_{\tau}X_{t}Y_{t}|\equiv\langle|\Delta_{\tau}X_{t}Y_{t}|\rangle$
of detrended profiles $\\{X_{t}\\}$ and $\\{Y_{t}\\}$ with time resolution
$\nu$ and $t=\nu,2\nu,...,\nu\lfloor\frac{T}{\nu}\rfloor$, where
$\lfloor\rfloor$ is a lower integer sign. We denote
$T^{\ast}=\nu\lfloor\frac{T}{\nu}\rfloor$, which varies with $\nu$, and we
write the $\tau$-lag difference as $\Delta_{\tau}X_{t}\equiv X_{t+\tau}-X_{t}$
and $\Delta_{\tau}X_{t}Y_{t}\equiv\Delta_{\tau}X_{t}\Delta_{\tau}Y_{t}$ for
better legibility. The covariance function scales as
$K_{xy,2}(\tau)\propto\tau^{H_{xy}}$ for the power-law cross-correlated
processes. Di Matteo et al. [65, 66, 67] suggest using the jackknife method to
obtain more precise estimates of the Hurst exponent. In our simulations, we
set $\tau_{min}=1$ and vary $\tau_{max}=5,\ldots,20$ to get the estimated
$H_{xy}$ as an average of these.

The detrended moving-average cross-correlation analysis (DMCA) [7, 53] is a
generalization of the detrending moving average (DMA) [68, 69]. The method is
in a way similar to DCCA as it assumes the power-law scaling of detrended
covariances. However, there is no box-splitting involved in the procedure,
which makes DMCA much less computationally demanding. Specifically, the
profiles $\\{X_{t}\\}$ and $\\{Y_{t}\\}$ are detrended using the moving
average of length $\kappa$ and the covariances of residuals
$F_{DMCA}^{2}(\kappa)$ scale as $F_{DMCA}^{2}(\kappa)\propto\kappa^{2H_{xy}}$
for the power-law cross-correlated processes. In our setting, we use the non-
weighted centered moving averages with $\kappa_{min}=11$ and
$\kappa_{max}=1+T/5$ with a step of 2, i.e. parallel to the DCCA setting.

###  2.2 Frequency domain estimators

Construction of the frequency domain estimators is based on the divergence at
origin of the cross power-spectrum of the power-law cross-correlated
processes. As the cross-power spectrum is unobservable, its estimation becomes
a crucial part of all the frequency domain procedures. For this purpose, a
cross-periodogram $I_{xy}(\lambda)$ is the simplest and the most popular tool.
It is defined as

$I_{xy}(\lambda_{j})=\frac{1}{2\pi}\sum_{k=-\infty}^{+\infty}{\widehat{\gamma}_{xy}(k)\exp(-i\lambda_{j}k)}=\frac{1}{2\pi
T}\sum_{t=1}^{T}{x_{t}\exp(-i\lambda_{j}t)}\sum_{t=1}^{T}{y_{t}\exp(i\lambda_{j}t)}=\\\
I_{x}(\lambda_{j})\overline{I_{y}(\lambda_{j})},$ (1)

where $T$ is the time series length, $\widehat{\gamma}_{xy}(k)$ is an
estimated cross-covariance at lag $k$, and $\lambda_{j}$ is a frequency
defined as $\lambda_{j}=2\pi j/T$, where $j=1,2,\ldots,\lfloor T/2\rfloor$ and
$\lfloor\rfloor$ is the nearest lower integer operator. The cross-periodogram
is thus defined between 0 and $\pi$ only. $I_{x}(\lambda_{j})$ is a
periodogram of series $\\{x_{t}\\}$, and $\overline{I_{y}(\lambda_{j})}$ is a
complex conjugate of a periodogram of series $\\{y_{t}\\}$. To overcome the
inconsistency of the raw periodogram in Eq. 1 [70], we use a smoothing
operator based on Daniell [71, 72]. The smoothed cross-periodogram is utilized
during the following estimation procedures.

Sela & Hurvich [25] propose the averaged periodogram estimator (APE) as a
bivariate generalization of the method of Robinson [73]. Using a cumulative
cross-periodogram

$\widehat{F_{xy}}(\lambda)=\frac{2\pi}{m}\sum_{j=1}^{\lfloor
m\lambda/2\pi\rfloor}{I_{xy}(\lambda_{j})},$ (2)

where $m\leq T/2$ is a bandwidth parameter and fixed $q\in(0,1)$, the
estimator is given as

$\widehat{H_{xy}}=1-\frac{\log\frac{\widehat{F_{xy}}(q\lambda_{m})}{\widehat{F_{xy}}(\lambda_{m})}}{2\log
q}.$ (3)

We follow the suggestion of Sela & Hurvich [25] and use $q=0.5$. Under
numerous assumptions, the estimator is consistent, which is supported by
simulations [25, 51].

The cross-periodogram estimator (XPE) proposed by Kristoufek [51] is based on
the power-law scaling of cross power-spectrum close to the origin with a
specific slope of $1-2H_{xy}$. The estimate of the bivariate Hurst exponent is
obtained from

$|I_{xy}(\lambda_{j})|\propto\lambda_{j}^{1-2H_{xy}}.$ (4)

The scaling is expected to hold only for $\lambda\rightarrow 0+$ so that the
regression is not performed over all frequencies. By choosing
$\lambda_{j}=2\pi j/T$ for $j=1,2,\ldots,m$ where $m\leq T/2$, the bivariate
Hurst exponent is estimated using only the information up to the selected
frequency specified by the bandwidth parameter $m$. The selection of $m$
influences the statistical properties of the estimator as an inclusion of more
frequencies lowers variance but increases bias of the estimator. For the
presented simulations, we follow the results of Kristoufek [51] and use
$m=T/10$ for XPE as well as for the last analyzed estimator.

Kristoufek [51] introduces the local cross-Whittle estimator (LXW) as a
bivariate generalization of the local Whittle estimator [74] which is a semi-
parametric maximum likelihood estimator based on the penalty function of
Künsch [75]. Specifically, the estimate of $H_{xy}$ is given as

$\widehat{H_{xy}}=\arg\min_{\frac{1}{2}<H_{xy}\leq 1}R(H_{xy}),$ (5)

where

$R(H_{xy})=\log\left(\frac{1}{m}\sum_{j=1}^{m}{\lambda_{j}^{2H_{xy}-1}|I_{xy}(\lambda_{j})|}\right)-\frac{2H_{xy}-1}{m}\sum_{j=1}^{m}{\log\lambda_{j}}$
(6)

and $\lambda_{j}=2\pi j/T$.

###  2.3 Simulations setting

We are interested in the effect of heavy tails on the power-law cross-
correlations estimators performance. As a representative of heavy-tailed
distributions, we select the $\alpha$-stable distributions which provide a
rich parametric flexibility. We use the parametrization of Nolan [76]. The
$\alpha$-stable distributions are characterized by four parameters – $\alpha$,
$\beta$, $\gamma$ and $\delta$. The heaviness of tails is represented by the
characteristic exponent $0<\alpha\leq 2$. Skewness is specified by parameter
$-1\leq\beta\leq 1$ with $\beta=0$ signifying symmetry, $\beta>0$ positive
skewness and $\beta<0$ negative skewness. Parameters $\gamma$ and $\delta$
describe scale and location of the distribution. The $\alpha$-stable
distributed random variable with $S(\alpha,\beta,\gamma,\delta)$ has the
following characteristic function:

$\footnotesize\phi(u)=\left\\{\begin{array}[]{lr}\exp(-\gamma^{\alpha}|u|^{\alpha}[1+i\beta(\tan\frac{\pi\alpha}{2})($sign$u)(|\gamma
u|^{1-\alpha}-1)]+i\delta u)&\alpha\neq 1\\\
\exp(-\gamma|u|[1+i\beta\frac{2}{\pi}($sign$u)\ln(\gamma|u|)]+i\delta
u)&\alpha=1\end{array}\right.$ (7)

There are only three cases with a closed-form density – Gaussian ($\alpha=2$,
$\beta=0$), Cauchy ($\alpha=1$, $\beta=0$), and Lévy ($\alpha=1/2$, $\beta=1$)
distributions. Importantly, the characteristic parameter $\alpha$ also
specifies the existing moments of the distribution. For $\alpha$-stable
distributions, only moments below $\alpha$ exist. Therefore, we have
nonexistent variance for $\alpha<2$ and even nonexistent mean for $\alpha<1$.

In the simulations, we focus on the effect of the characteristic parameter
$\alpha$, which manipulates the distribution tails, on performance of six
estimators of power-law cross-correlations. To do so, we follow two settings.
For the first setting, one of the processes is the standard Gaussian noise and
the other is the $\alpha$-stable random variable. For the latter process, we
vary the parameter $\alpha$ between 1.1 and 2 with a step of 0.1. The other
parameters are fixed to $\beta=0$, $\gamma=\sqrt{2}/2$ and $\delta=0$. The
simulated series are standardized to have a unit variance. For the second
setting, both processes are $\alpha$-stable distributed with the same $\alpha$
parameter. For each parameter setting, we simulate 1000 series with time
series lengths of $T=500,1000,5000,10000$ to see how the statistical
properties of the estimators depend on the number of observations. The studied
series are independent so that the expected bivariate Hurst exponent is equal
to 0.5. As measures of performance, we study bias, standard deviation and mean
squared errors of the estimators.

##  3 Results and Discussion

We start the results description by examination of bias of the estimators. In
Figs. 1 and 4, we present the mean values (full line), and the 2.5th and
97.5th quantiles (dashed line) based on 1000 simulations for each setting. The
darker the line, the higher the time series length ($T=500,1000,5000,10000$).
The red line stands for the expected value of $H_{xy}$ for the given setting,
i.e. $H_{xy}=0.5$ for all specifications. Fig. 1 shows the results for the
setting with one Gaussian and one $\alpha$-stable process, Fig. 4 represents
two $\alpha$-stable processes with same $\alpha$. The outcome is very
straightforward. The frequency domain estimators are unbiased for all settings
of $\alpha$ and even the quantiles are very stable for different
characteristic exponents. These estimators are thus practically untouched by
heavy tails of the underlying series. The confidence bands become narrower
with the increasing time series length, which is desirable. This is true for
both settings. The situation differs for the time domain estimators. For low
levels of $\alpha$, i.e. for heavy tails, the estimators are biased upwards.
This could result in spuriously reported cross-persistence. The bias decreases
with an increasing $\alpha$ and it vanishes for $\alpha=2$ (normal tails).
Interestingly, the confidence bands of the estimates get narrower with lighter
tails and even for some cases of heavy tails (mainly for shorter series), the
bands are narrower than the ones of the frequency domain estimators. The
situation is even more pronounces for the setting with two heavy-tailed
distributions. For very heavy tails, there are situations when the confidence
bands climb above the reasonable value of $H_{xy}=1$.

To further examine this discrepancy, Figs. 2 and 5 report the estimator
standard deviation dependence on parameter $\alpha$ for all investigated
estimators. The results obviously reflect the results for confidence bands but
these are more transparent here (the figures are split based on the specific
time series lengths for an easier comparison between estimators). In fact,
standard deviation of the frequency domain estimators is practically
independent of $\alpha$. The highest standard deviation is reported for APE
followed by XPE and LXW with quite similar results. For short time series with
$T=500$ and $T=1000$, the time domain estimators possess lower variance for
practically all levels of $\alpha$. The lowest variance is found for HXA
followed by DCCA and DMCA, respectively. The evidence gets more mixed for
$T=5000$. And for $T=10000$, DCCA and DMCA estimators are dominated by the
frequency domain estimators for all levels of $\alpha$ but the Gaussian case.
The HXA estimator outperforms even the frequency domain estimators for
$\alpha>1.5$. These implications are valid for the case of one Gaussian and
one $\alpha$-stable distribution in the simulations (Fig. 2). However, the
situation gets more clearcut for the case of two $\alpha$-stable distributions
(Fig. 5). Standard deviation of the frequency domain estimators is again
reasonably stable across different $\alpha$. Interestingly, their standard
deviation even slightly increases with lighter tails (increasing $\alpha$).
The general level of the time domain estimators standard deviations increases
markedly (compare Figs. 2 and 5). Now even for short series, the tails need to
be quite light ($\alpha>1.5$ for $T=500$ and $\alpha>1.7$ for $T=1000$) for
the time domain estimators to clearly dominate. For the longer series with
$T=5000$ and $T=10000$, only HXA competes with the frequency domain estimators
and it outperforms them only for $\alpha>1.6$ and $\alpha>1.7$ for $T=5000$
and $T=10000$, respectively. For the frequency domain estimators, it again
holds that XPE and LXW are the best performing estimators while LXW has a
slight edge over XPE.

A closer look at both Figs. 2 and 5 uncover a clear difference between the
behavior of standard deviations with respect to time series length for both
groups of estimators. For the frequency domain estimators, we can easily
observe that their standard deviation decays approximately at a rate of
$1/\sqrt{T}$ (or $1/T$ for variance). However, the decay is much slower for
the time domain estimators (at approximately $T^{-0.15}$ for standard
deviation). This is a reflection of the univariate frequency domain estimators
being asymptotically normal and efficient [73, 74] whereas no such quality is
known for the time domain estimators.

The tradeoff between bias and variance is nicely summed together in the mean
squared error (MSE), which comprises both. In Figs. 3 and 6, we compare MSE
for all estimators and different parameter settings. We start with case of one
Gaussian and one $\alpha$-stable process. For short time series with $T=500$,
the time domain estimators dominate the frequency domain ones for
$\alpha>1.3$. The methods are tied for the lower values of $\alpha$. HXA and
DCCA come out as winners with HXA having a slight edge over DCCA for light
tails ($\alpha>1.6$). The breaking point increases from $\alpha\approx 1.3$ to
$\alpha\approx 1.5$ for $T=1000$ but the results remain qualitatively
unchanged. For the longer time series lengths ($T=5000$ and $T=10000$), the
frequency domain methods dominate for practically all levels of $\alpha$ apart
from light tails with $\alpha\geq 1.8$ where HXA gives more precise estimates
in the MSE sense. The other time domain methods do not provide such results
for long time series. For the frequency domain estimators, the ranking based
on standard deviations translates perfectly to the mean squared error case.
Specifically, LXW is the best performing frequency domain estimator tightly
followed by XPE. The APE estimator margin is evident. The results are again
more pronounced for the case of two $\alpha$-stable distributions. In Fig. 6,
we can see that the breaking points between better performance of the
frequency domain estimators and time domain estimators move up. Specifically,
we have the breaking point of $\alpha\approx 1.6$ for $T=500$ and
$\alpha\approx 1.7$ for $T=1000$. For the longer series ($T=5000$ and
$T=10000$), again only HXA can compete with APE, LXW and XPE and it dominates
them only for $\alpha>1.9$. The ranking of the frequency domain estimators
remains unchanged compared to the other setting.

The presented results provide clear implications. First, the frequency domain
estimators (APE, LXW and XPE) are not affected by heavy tails. Second, the
time domain estimators (DCCA, DMCA and HXA) are biased upwards by heavy tails.
Third, the time domain estimators have lower variance than the frequency
domain ones for shorter time series ($T=500$ and $T=1000$) and lighter tails
(above approximately $\alpha=1.5$). Putting the findings together to provide
some recommendations for use of the estimators, we suggest using the time
domain estimators for short time series and light tails close to the normal
ones. Otherwise, we recommend to use the frequency domain estimators,
preferably LXW and XPE. Comparison of two different settings of having either
one Gaussian and one $\alpha$-stable distributed process or two
$\alpha$-stable distributed ones only highlights the sensitivity of the time
domain estimators to heavy tails.

The findings also have important implications for the stream of literature
primarily using the time domain estimators. Combined with the results of
Barunik & Kristoufek [36] who show that the univariate versions of DCCA, DMCA
and HXA (detrended fluctuation analysis, DFA, detrending moving average, DMA,
and height-height correlation analysis, HCCA, respectively) are unbiased for
various levels of heavy tails, we provide a possible explanation for the
frequently reported estimated bivariate Hurst exponent $\widehat{H}_{xy}$
being higher than the average of the separate estimated Hurst exponents
$\widehat{H}_{x}$ and $\widehat{H}_{y}$, i.e.
$\widehat{H}_{xy}>\frac{\widehat{H}_{x}+\widehat{H}_{y}}{2}$. As the Hurst
exponent estimation is unbiased by heavy tails in the univariate setting, the
upward bias in the bivariate setting for heavy tails implies the possibility
of $\widehat{H}_{xy}>\frac{\widehat{H}_{x}+\widehat{H}_{y}}{2}$. However, the
recent study of Kristoufek [27] analytically shows that
$H_{xy}\leq\frac{H_{x}+H_{y}}{2}$ for very generally defined processes. In
addition, the author argues that one of the possible reasons for reporting
$\widehat{H}_{xy}>\frac{\widehat{H}_{x}+\widehat{H}_{y}}{2}$ is a finite
sample bias. Here we add the bias due to distributional properties of the
underlying processes, namely the heavy tails. Estimation of the bivariate
Hurst exponent and related separate Hurst exponents is thus a complex matter
which needs to be approached and treated accordingly.

The results bring additional challenges to the time domain estimators
construction, specifically its application under heavy tails. As the
procedures are based on the absolute values of covariance scaling, the methods
can confuse the scaling of covariances with scaling of tails of the separate
processes. As shown by Barunik & Kristoufek [36], the time domain estimators
are unbiased under heavy tails when estimating long-range dependence. When the
setting is extended to the absolute values scaling, it very well corresponds
to the $\alpha$ parameter of the $\alpha$-stable distributions. However, this
could create a problem as the procedures might mistake the covariance absolute
value scaling for the scaling of absolutes values of the separate processes,
hence the upward bias reported in the simulation study. The remedy does not
necessarily need to be too complicated as the heavy tails bias can be
estimated using shuffling procedures or bootstrapping. This way, the time
domain estimators could successfully compete with the frequency domain
estimators even under heavy tails.

## Acknowledgements

The research leading to these results has received funding from the Czech
Science Foundation under project No. 14-11402P.

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Figure 1: Mean values of the estimators for different tail exponents (Setting I). The solid lines represent the average values of 1000 simulations for the given setting. The dashed lines show the 95% confidence intervals, i.e. the 2.5th and the 97.5th quantiles. The $x$-axis gives values of the tail exponent of series $\\{y_{t}\\}$ simulated as the $\alpha$-stable distributed process with the characteristic exponent $\alpha$, i.e. the more to the left of the axis the heavier the tails of the underlying process. The $\\{x_{t}\\}$ process is the standard Gaussian noise. Both processes are standardized to have a unit variance. The shades of grey stand for the different time series lengths, $T=500,1000,5000,10000$, the darker the color the longer the series. The red line stands for the theoretical value of $H_{xy}=0.5$. On the one hand, the frequency domain estimators are unbiased whereas the time domain estimators are biased under heavy tails. On the other hand, the latter ones have narrower confidence intervals (especially for HXA). This is further studied in Fig. 2 and 3.  |
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Figure 2: Standard deviation of the estimators for different tail exponents (Setting I). The figures show the dependence of estimators’ standard deviation on the tail exponent as well as the time series length. The variance is very stable for the frequency domain estimators. For the time domain estimators, the variance depends on the tail exponent strongly. For the shorter series, i.e. $T=500,1000$, the variance of the latter group is much lower compared to the former group. The differences shrink considerably for the longer series, i.e. $T=5000,10000$, where the results become mixed and more in favor of the frequency domain estimators.  |
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Figure 3: Mean squared error of the estimators for different tail exponents (Setting I). Mean squared error combines the performance of the estimators based on their bias as well as their variance. MSE is stable for the frequency domain estimators but strongly dependent on the tail exponent for the time domain estimators. The discrepancy between the two groups is dependent both on the tail index and on the time series length. For the shorter series, i.e. $T=500,1000$, the latter group outperforms the former for most tail index values. For the longer series, the former group dominates apart from tails close to normality where the HXA method strongly dominates all the other methods.  |
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Figure 4: Mean values of the estimators for different tail exponents (Setting II). The solid lines represent the average values of 1000 simulations for the given setting. The dashed lines show the 95% confidence intervals, i.e. the 2.5th and the 97.5th quantiles. The $x$-axis gives values of the tail exponent of series $\\{x_{t}\\}$ and $\\{y_{t}\\}$, which are both simulated as the $\alpha$-stable distributed process with the characteristic exponent $\alpha$, i.e. the more to the left of the axis the heavier the tails of the underlying process. The shades of grey stand for the different time series lengths, $T=500,1000,5000,10000$, the darker the color the longer the series. The red line stands for the theoretical value of $H_{xy}=0.5$. The results are qualitatively very similar to Setting I illustrated in Fig. 1 with differences being more pronounced. This is further studied in Fig. 5 and 6.  |
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Figure 5: Standard deviation of the estimators for different tail exponents (Setting II). The figures show the dependence of estimators’ standard deviation on the tail exponent as well as the time series length. The variance is again very stable for the frequency domain estimators. It even slightly increasing with lighter tails. For the time domain estimators, the variance depends on the tail exponent strongly. For the shorter series, i.e. $T=500,1000$, and lighter tails, i.e. approximately $\alpha>1.5$, the variance of the latter group is much lower compared to the former group. The differences shrink considerably for the longer series, i.e. $T=5000,10000$, where the frequency domain estimators strongly dominate for the most levels of tails.  |
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Figure 6: Mean squared error of the estimators for different tail exponents
(Setting II). Mean squared error combines the performance of the estimators
based on their bias as well as their variance. The results are qualitatively
very similar to Fig. 3. MSE is again stable for the frequency domain
estimators but strongly dependent on the tail exponent for the time domain
estimators. The discrepancy between the two groups is dependent both on the
tail index and on the time series length. The results are in hand with the
standard deviation behavior in Fig. 5.