The origin of the concept of ordinal patterns is in the theory of dynamical systems. The idea is to consider the order of the values within a data vector instead of the full metrical information. The ordinal information is encoded as a permutation (cf. Section 3). Already in the first papers on the subject, the authors considered entropy concepts related to this ordinal structure (cf. [1]). There is an interesting relationship between these concepts and the well-known Komogorov–Sinai entropy (cf. [2,3]). Additionally, an ordinal version of the Feigenbaum diagram has been dealt with e.g., in [4]. In [5], ordinal patterns were used in order to estimate the Hurst parameter in long-range dependent time series. Furthermore, Ref. [6] have proposed a test for independence between time series (cf. also [7]). Hence, the concept made its way into the area of statistics. Instead of long patterns (or even letting the pattern length tend to infinity), rather short patterns have been considered in this new framework. Furthermore, ordinal patterns have been used in the context of ARMA processes [8] and change-point detection within one time series [9]. In [10], ordinal patterns were used for the first time in order to analyze the dependence between two time series. Limit theorems for this new concept were proven in a short-range dependent framework in [11]. Ordinal pattern dependence is a promising tool, which has already been used in financial, biological and hydrological data sets, see in this context, also [12] for an analysis of the co-movement of time series focusing on symbols. In particular, in the context of hydrology, the data sets are known to be long-range dependent. Therefore, it is important to also have limit theorems available in this framework. We close this gap in the present article.

All of the results presented in this article have been established in the Ph.D. thesis of I. Nüßgen written under the supervision of A. Schnurr.

The article is structured as follows: in the subsequent section, we provide the reader with the mathematical framework. The focus is on (multivariate) long-range dependence. In Section 3, we recall the concept of ordinal pattern dependence and prove our main results. We present a simulation study in Section 4 and close the paper by a short outlook in Section 5.

We consider a stationary d-dimensional Gaussian time series ${\left(Y_j\right)}_{j\in \mathbb{Z}}$ (for $d\in \mathbb{N}$), with: such that $\mathbb{E}\left(Y_j^{\left(p\right)}\right)=0$ and $\mathbb{E}\left({\left(Y_j^{\left(p\right)}\right)}^2\right)=1$ for all $j\in \mathbb{Z}$ and $p=1,\dots ,d$. Furthermore, we require the cross-correlation function to fulfil $\left|r^{(p,q)}\left(k\right)\right|<1$ for $p,q=1,\dots ,d$ and $k\ge 1$, where the component-wise cross-correlation functions $r^{(p,q)}\left(k\right)$ are given by $r^{(p,q)}\left(k\right)=\mathbb{E}\left(Y_j^{\left(p\right)}{Y}_{j+k}^{\left(q\right)}\right)$ for each $p,q=1,\dots ,d$ and $k\in \mathbb{Z}$. For each random vector $Y_j$, we denote the covariance matrix by ${\mathsf{\Sigma }}_d$, since it is independent of j due to stationarity. Therefore, we have ${\mathsf{\Sigma }}_d={\left(r^{(p,q)}\left(0\right)\right)}_{p,q=1,\dots ,d}$.

We specify the dependence structure of ${\left(Y_j\right)}_{j\in \mathbb{Z}}$ and turn to long-range dependence: we assume that for the cross-correlation function $r^{(p,q)}\left(k\right)$ for each $p,q=1,\dots ,d$, it holds that: with $L_{p,q}\left(k\right)\to L_{p,q}$$(k\to \infty )$ for finite constants $L_{p,q}\in [0,\infty )$ with $L_{p,p}\ne 0$, where the matrix $L={\left(L_{p,q}\right)}_{p,q=1,\dots ,d}$ has full rank, is symmetric and positive definite. Furthermore, the parameters $d_p,d_q\in \left(0,\frac{1}{2}\right)$ are called long-range dependence parameters. Therefore, ${\left(Y_j\right)}_{j\in \mathbb{Z}}$ is multivariate long-range dependent in the sense of [13], Definition 2.1.

The processes we want to consider have a particular structure, namely for $h\in \mathbb{N}$, we obtain for fixed $j\in \mathbb{Z}$:

The following relation holds between the extendend process ${\left(Y_{j,h}\right)}_{j\in \mathbb{Z}}$ and the primarily regarded process ${\left(Y_j\right)}_{j\in \mathbb{Z}}$. For all $k=1,\dots ,dh$, $j\in \mathbb{Z}$ we have: where $\lfloor x\rfloor =max\{k\in \mathbb{Z}:k\le x\}$. Note that the process ${\left(Y_{j,h}\right)}_{j\in \mathbb{Z}}$ is still a centered Gaussian process since all finite-dimensional marginals of ${\left(Y_j\right)}_{j\in \mathbb{Z}}$ follow a normal distribution. Stationarity is also preserved since for all $p,q=1,\dots ,dh$, $p\le q$ and $k\in \mathbb{Z}$, the cross-correlation function $r^{(p,q,h)}\left(k\right)$ of the process ${\left(Y_{j,h}\right)}_{j\in \mathbb{Z}}$ is given by and the last line does not depend on j. The covariance matrix ${\mathsf{\Sigma }}_{d,h}$ of $Y_{j,h}$ has the following structure:

Hence, we arrive at: where ${\mathsf{\Sigma }}_h^{(p,q)}=\mathbb{E}\left({\left(Y_1^{\left(p\right)},\dots ,Y_h^{\left(p\right)}\right)}^t\left(Y_1^{\left(q\right)},\dots ,Y_h^{\left(q\right)}\right)\right)={\left(r^{(p,q)}(i-k)\right)}_{1\le i,k\le h}$, $p,q=1,\dots ,d$. Note that ${\mathsf{\Sigma }}_h^{(p,q)}\in {\mathbb{R}}^{h\times h}$ and $r^{(p,q)}\left(k\right)=r^{(q,p)}(-k)$, $k\in \mathbb{Z}$ since we are studying cross-correlation functions.

Therefore, we finally have to show that based on the assumptions on ${\left(Y_j\right)}_{j\in \mathbb{Z}}$, the extended process is still long-range dependent.

Hence, we have to consider the cross-correlations again: since $p^{*},q^{*}\in \{1,\dots ,d\}$ and $m^{*}\in \{0,\dots ,h-1\}$, with $p^{*}:=⌊\frac{p-1}{h}⌋+1$, $q^{*}:=⌊\frac{q-1}{h}⌋+1$ and $m^{*}=(q-p)modh$.

Let us remark that $a_k\simeq b_k\iff {lim}_{k\to \infty }\frac{a_k}{b_k}=1$.

Therefore, we are still dealing with a multivariate long-range dependent Gaussian process. We see in the proofs of the following limit theorems that the crucial parameters that determine the asymptotic distribution are the long-range dependence parameters $d_p$, $p=1,\dots ,d$ of the original process ${\left(Y_j\right)}_{j\in \mathbb{Z}}$ and therefore, we omit the detailed description of the parameters $d_{p^{*}}$ herein.

It is important to remark that the extended process ${\left(Y_{j,h}\right)}_{j\in \mathbb{Z}}$ is also long-range dependent in the sense of [14], p. 2259, since: with: and $L\left(k\right)$ can be chosen as any constant $L_{p,q}$ that is not equal to zero, so for simplicity, we assume without a loss of generality $L_{1,1}\ne 0$, and therefore, $L\left(k\right)=L_{1,1}$, since the condition in [14] only requires convergence to a finite constant $b_{p^{*},q^{*}}$. Hence, we may apply the results in [14] in the subsequent results.

We define the following set, which is needed in the proofs of the theorems of this section. and denote the corresponding long-range dependence parameter to each $p\in P^{*}$ by

We briefly recall the concept of Hermite polynomials as they play a crucial role in determining the limit distribution of functionals of multivariate Gaussian processes.

Their multivariate extension is given by the subsequent definition.

Let us remark that the case $k=(0,\dots ,0)$ is excluded here due to the assumption $\mathbb{E}\left(f\left(X\right)\right)=0$.

Analogously to the univariate case, the family of multivariate Hermite polynomials

$\left\{H_{k_1,\dots ,k_d},k_1,\dots ,k_d\in \mathbb{N}\right\}$ forms an orthogonal basis of $L^2\left({\mathbb{R}}^d,{\phi }_{I_d}\right)$, which is defined as

The parameter ${\phi }_{I_d}$ denotes the density of the d-dimensional standard normal distribution, which is already divided into the product of the univariate densities $\phi$ in the formula above.

We denote the Hermite coefficients by

The Hermite rank $m\left(f,I_d\right)$ of f with respect to the distribution $\mathcal{N}\left(0,I_d\right)$ is defined as the largest integer m, such that:

Having these preparatory results in mind, we derive the multivariate Hermite expansion given by

We focus on the limit theorems for functionals with Hermite rank 2. First, we introduce the matrix-valued Rosenblatt process. This plays a crucial role in the asymptotics of functionals with Hermite rank 2 applied to multivariate long-range dependent Gaussian processes. We begin with the definition of a multivariate Hermitian–Gaussian random measure $\tilde{B}\left(\mathrm{d}\lambda \right)$ with independent entries given by where ${\tilde{B}}^{\left(p\right)}\left(\mathrm{d}\lambda \right)$ is a univariate Hermitian–Gaussian random measure as defined in [16], Definition B.1.3. The multivariate Hermitian–Gaussian random measure $\tilde{B}\left(\mathrm{d}\lambda \right)$ satisfies: and: where $\tilde{B}{\left(\mathrm{d}\lambda \right)}^{*}=\left(\overline{B^{\left(1\right)}\left(\mathrm{d}\lambda \right)},\dots ,\overline{B^{\left(d\right)}\left(\mathrm{d}\lambda \right)}\right)$ denotes the Hermitian transpose of $\tilde{B}\left(\mathrm{d}\lambda \right)$. Thus, following [14], Theorem 6, we can state the spectral representation of the matrix-valued Rosenblatt process $Z_{2,H}\left(t\right)$, $t\in [0,1]$ as where each entry of the matrix is given by

The double prime in ${\int }_{{\mathbb{R}}^2}^{″}$ excludes the diagonals $\left|{\lambda }_i\right|=\left|{\lambda }_j\right|$, $i\ne j$ in the integration. For details on multiple Wiener-Itô integrals, as can be seen in [17].

The following results were taken from [18], Section 3.2. The corresponding proofs were outsourced to the Appendix A.

It is possible to soften the assumptions in Theorem 1 to allow for mixed cases of short- and long-range dependence.

However, with a mild technical assumption on the covariances of the one-dimensional marginal Gaussian processes that is often fulfilled in applications, there is another way of normalizing the partial sum on the right-hand side in Theorem 1, this time explicitly for the case $\#P^{*}=2$ and $h\in \mathbb{N}$, such that the limit can be expressed in terms of two standard Rosenblatt random variables. This yields the possibility of further studying the dependence structure between these two random variables. In the following theorem, we assume $\#P^{*}=d=2$ for the reader’s convenience.

Ordinal pattern dependence is a multivariate dependence measure that compares the co-movement of two time series based on the ordinal information. First introduced in [10] to analyze financial time series, a mathematical framework including structural breaks and limit theorems for functionals of absolutely regular processes has been built in [11]. In [19], the authors have used the so-called symbolic correlation integral in order to detect the dependence between the components of a multivariate time series. Their considerations focusing on testing independence between two time series are also based on ordinal patterns. They provide limit theorems in the i.i.d.-case and otherwise use bootstrap methods. In contrast, in the mathematical model in the present article, we focus on asymptotic distributions of an estimator of ordinal pattern dependence having a bivariate Gaussian time series in the background but allowing for several dependence structures to arise. As it will turn out in the following, this yields central but also non-central limit theorems.

We start with the definition of an ordinal pattern and the basic mathematical framework that we need to build up the ordinal model.

Let $S_h$ denote the set of permutations in $\{0,\dots ,h\}$, $h\in {\mathbb{N}}_0$ that we express as $(h+1)$-dimensional tuples, assuring that each tuple contains each of the numbers above exactly once. In mathematical terms, this yields: as can be seen in [11], Section 2.1.

The number of permutations in $S_h$ is given by $\#S_h=(h+1)!$. In order to get a better intuitive understanding of the concept of ordinal patterns, we have a closer look at the following example, before turning to the formal definition.

Formally, the aforementioned procedure can be defined as follows, as can be seen in [11], Section 2.1.

The last condition assures the uniqueness of $\pi$ if there are ties in the data sets. In particular, this condition is necessary if real-world data are to be considered.

In Figure 2, all ordinal patterns of length $h=2$ are shown. As already mentioned in the introduction, from the practical point of view, a highly desirable property of ordinal patterns is that they are not affected by monotone transformations, as can be seen in [5], p. 1783.

Mathematically, this means that if $f:\mathbb{R}\to \mathbb{R}$ is strictly monotone, then:

In particular, this includes linear transformations $f\left(x\right)=ax+b$, with $a\in {\mathbb{R}}^{+}$ and $b\in \mathbb{R}$.

Following [11], Section 1, the minimal requirement of the data sets we use for ordinal analysis in the time series context, i.e., for ordinal pattern probabilities as well as for ordinal pattern dependence later on, is ordinal pattern stationarity (of order h). This property implies that the probability of observing a certain ordinal pattern of length h remains the same when shifting the moving window of length h through the entire time series and is not depending on the specific points in time. In the course of this work, the time series, in which the ordinal patterns occur, always have either stationary increments or are even stationary themselves. Note that both properties imply ordinal pattern stationarity. The reason why requiring stationary increments is a sufficient condition is given in the following explanation.

One fundamental property of ordinal patterns is that they are uniquely determined by the increments of the considered time series. As one can imagine in Example 1, the knowledge of the increments between the data points is sufficient to obtain the corresponding ordinal pattern. In mathematical terms, we can define another mapping $\tilde{\mathsf{\Pi }}$, which assigns the corresponding ordinal pattern to each vector of increments, as can be seen in [5], p. 1783.

We define the two mappings, following [5], p. 1784:

An illustrative understanding of these mappings is given as follows. The mapping $\mathcal{S}\left(\pi \right)$, which is the spatial reversion of the pattern $\pi$, is the reflection of $\pi$ on a horizontal line, while $\mathcal{T}\left(\pi \right)$, the time reversal of $\pi$, is its reflection on a vertical line, as one can observe in Figure 3.

Based on the spatial reversion, we define a possibility to divide $S_h$ into two disjoint sets.

Note that this definition does not yield the uniqueness of $S_h^{*}$.

We stick to the formal definition of ordinal pattern dependence, as it is proposed in [11], Section 2.1. The considered moving window consists of $h+1$ data points, and hence, h increments. We define: and:

Then, we define ordinal pattern dependence $OPD$ as

The parameter q represents the hypothetical case of independence between the two time series. In this case, p and q would obtain equal values and therefore, $OPD$ would equal zero. Regarding the other extreme, the case in which both processes coincide or one is a strictly monotone increasing transform of the other one, we obtain the value 1. However, in the following, we assume $p\in (0,1)$ and $q\in (0,1)$.

Note that the definition of ordinal pattern dependence in (17) only measures positive dependence. This is no restriction in practice, because negative dependence can be investigated in an analogous way, by considering $OP{D}_{X^{\left(1\right)},-X^{\left(2\right)}}$. If one is interested in both types of dependence simultaneously, in [11], the authors propose to use ${\left(OP{D}_{X^{\left(1\right)},X^{\left(2\right)}}\right)}_{+}-{\left(OP{D}_{X^{\left(1\right)},-X^{\left(2\right)}}\right)}_{+}$. To keep the notation simple, we focus on $OPD$ as it is defined in (17).

We compare whether the ordinal patterns in ${\left(X_j^{\left(1\right)}\right)}_{j\in \mathbb{Z}}$ coincide with the ones in ${\left(X_j^{\left(2\right)}\right)}_{j\in \mathbb{Z}}$. Recall that it is an essential property of ordinal patterns that they are uniquely determined by the increment process. Therefore, we have to consider the increment processes ${\left(Y_j\right)}_{j\in \mathbb{Z}}={\left(\left(Y_j^{\left(1\right)},Y_j^{\left(2\right)}\right)\right)}_{j\in \mathbb{Z}}$ as defined in (1) for $d=2$, where $Y_j^{\left(p\right)}=X_j^{\left(p\right)}-X_{j-1}^{\left(p\right)}$, $p=1,2$. Hence, we can also express p and q (and consequently $OPD$) as a probability that only depends on the increments of the considered vectors of the time series. Recall the definition of ${\left(Y_{j,h}\right)}_{j\in \mathbb{Z}}$ for $d=2$, given by such that $Y_{j,h}\sim \mathcal{N}\left(0,{\mathsf{\Sigma }}_{2,h}\right)$ with ${\mathsf{\Sigma }}_{2,h}$ as given in (6).

In the course of this article, we focus on the estimation of p. For a detailed investigation of the limit theorems for estimators of $OPD$, we refer to [18]. We define the estimator of p, the probability of coincident patterns in both time series in a moving window of fixed length, by where:

Figure 4 illustrates the way ordinal pattern dependence is estimated by ${\widehat{p}}_n$. The patterns of interest that are compared in each moving window are colored in red.

Having emphasized the crucial importance of the increments, we define the following conditions on the increment process ${\left(Y_j\right)}_{j\in \mathbb{Z}}$: let ${\left(Y_j\right)}_{j\in \mathbb{Z}}$ be a bivariate, stationary Gaussian process with $Y_j^{\left(p\right)}\sim \mathcal{N}(0,1)$, $p=1,2$:

Furthermore, in both cases, it holds that $\left|r^{(p,q)}\left(k\right)\right|<1$ for $p,q=1,2$ and $k\ge 1$ to exclude ties.

We begin with the investigation of the asymptotics of ${\widehat{p}}_n$. First, we calculate the Hermite rank of ${\widehat{p}}_n$, since the Hermite rank determines for which ranges of $d^{*}$ the estimator ${\widehat{p}}_n$ is still long-range dependent. Depending on this range, different limit theorems may hold.