arXiv subjects

Alexander Schnurr

Publications and source records attributed to Alexander Schnurr.

At least 19 recordsLinked to original sources

arXiv2025

Limit Theorems for the Symbolic Correlation Integral and the Renyi-2 Entropy under Short-range Dependence

The symbolic correlation integral provides a way to measure the complexity of time series and dynamical systems. In the present article we prove limit results for an estimator of this quantity which is based on U-statistics under the assumption of short-range dependence. To this end, we slightly generalize classical limit results in the framework of 1-approximating functionals. Furthermore, we carefully analyze the limit variance. A simulation study with ARMA and ARCH time series as well as a real world data example are also provided. In the latter we show how our method could be used to analyze EEG data in the context of epileptic seizures.

math.ST↗

arXiv2025

Depth Patterns and their Applications in Animal Tracking

We establish a definition of ordinal patterns for multivariate data sets based on the concept of Tukey's halfspace depth. Given the definition of these \emph{depth patterns}, we are interested in the probabilities of observing specific patterns in time series. For this, we consider the relative frequency of depth patterns as natural estimators for their occurrence probabilities. Depending on the choice of reference distribution and the relation between reference and data distribution, we distinguish different settings that are considered separately. Within these settings we study statistical properties of depth pattern probabilities, establishing consistency and asymptotic normality under the assumption of weakly dependent time series. Since our concept only depends on ordinal depth information, the resulting values are robust under small perturbations and measurement errors. We emphasize the applicability of our method by analyzing the depth patterns which are found in seal pubs' movement. We use our approach in order to choose an appropriate model out of a range of two-dimensional random walks, which are commonly used in mathematical biology.

math.ST↗

arXiv2024

Ordinal pattern dependence and multivariate measures of dependence

Ordinal pattern dependence has been introduced in order to capture co-monotonic behavior between two time series. This concept has several features one would intuitively demand from a dependence measure. It was believed that ordinal pattern dependence satisfies the axioms which Grothe et al. (Journal of Multivariate Analysis 123, 2014) proclaimed for a multivariate measure of dependence. In the present article we show that this is not true and that there is a mistake in the article Betken et al. (Journal of Multivariate Analysis 186, 2021). Furthermore we show that ordinal pattern dependence satisfies a slightly modified set of axioms.

math.ST↗

arXiv2024

Fixing a Minor Mistake in the Theory of Stochastic Integration and Differential Equations

When considering stochastic integration and the theory of stochastic differential equations, P. Protter's textbook \cite{protter} undoubtedly is a main piece of standard literature. Not only is it well-written, but it also contains various profound results regarding these fields. Unfortunately, Theorem 12 of Chapter V, which presents an equivalence to uniform convergence on compacts in probability, is found to be incorrect. Given that numerous important results rely on this theorem, this paper aims to present a corrected version of it.

math.PR↗

arXiv2024

A Comparison of Different Representations of Ordinal Patterns and Their Usability in Data Analysis

We describe and analyze different approaches to represent ordinal patterns. All of these can be found in the literature. The most important representations (plus sub-classes) are compared in terms of their applicability from different angles. Namely we consider digital implementation, inverse patterns and ties between values. At the end we provide a guideline on which occasions which representation should be used.

cs.DB↗

arXiv2024

Generalized Ordinal Patterns Allowing for Ties and Their Applications in Hydrology

When using ordinal patterns, which describe the ordinal structure within a data vector, the problem of ties appeared permanently. So far, model classes were used which do not allow for ties; randomization has been another attempt to overcome this problem. Often, time periods with constant values even have been counted as times of monotone increase. To overcome this, a new approach is proposed: it explicitly allows for ties and, hence, considers more patterns than before. Ties are no longer seen as nuisance, but to carry valuable information. Limit theorems in the new framework are provided, both, for a single time series and for the dependence between two time series. The methods are used on hydrological data sets. It is common to distinguish five flood classes (plus 'absence of flood'). Considering data vectors of these classes at a certain gauge in a river basin, one will usually encounter several ties. Co-monotonic behavior between the data sets of two gauges (increasing, constant, decreasing) can be detected by the method as well as spatial patterns. Thus, it helps to analyze the strength of dependence between different gauges in an intuitive way. This knowledge can be used to asses risk and to plan future construction projects.

stat.AP↗

arXiv2024

Operator-stable-like Processes

In the present paper, we introduce so-called operator-stable-like processes. Roughly speaking, they behave locally like operator-stable processes, but they need not to be homogenous in space. Having shown existence for this class of processes, we analyze maximal estimates, the existence of moments, the short- and long-time behavior of the sample paths and $p$-variation. The class introduced here includes stable-like processes as special case.

math.PR↗

arXiv2023

On the rôle of singular functions in extending the probabilistic symbol to its most general class

The probabilistic symbol is the right-hand side derivative of the characteristic functions corresponding to the one-dimensional marginals of a stochastic process. This object, as long as the derivative exists, provides crucial information concerning the stochastic process. For a Lévy process, one obtains the characteristic exponent while the symbol of a (rich) Feller process coincides with the classical symbol which is well known from the theory of pseudodifferential operators. Leaving these classes behind, the most general class of processes for which the symbol still exists are Lévy-type processes. It has been an open question, whether further generalizations are possible within the framework of Markov processes. We answer this question in the present article: within the class of Hunt semimartingales, Lévy-type processes are exactly those for which the probabilistic symbol exists. Leaving quasi-continuity behind, one can construct processes admitting a symbol. However, we show, that the applicability of the symbol might be lost for these processes. Surprisingly, in our proofs the upper and lower Dini derivatives corresponding to certain singular functions play an important rôle.

math.PR↗

arXiv2023

The Time-Dependent Symbol of a Non-Homogeneous Itô Process and corresponding Maximal Inequalities

The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes we derive maximal inequalities which we apply to generalize the Blumenthal-Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample patsh, the existence of exponential moments and the finiteness of p-variation. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities.

math.PR↗

arXiv2022

From Markov Processes to Semimartingales

In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the `natural' class of processes for many concepts first developed in the Markovian framework. As an example, stochastic differential equations have been invented as a tool to study Markov processes but nowadays are treated separately in the literature. Moreover, the killing of processes has been known for decades before it made its way to the theory of semimartingales most recently. We describe, when these and other important concepts have been invented in the theory of Markov processes and how they were transferred to semimartingales. Further topics include the symbol, characteristics and generalizations of Blumenthal-Getoor indices. Some additional comments on relations between Markov processes and semimartingales round out the paper.

math.PR↗

arXiv2021

Ordinal pattern dependence as a multivariate dependence measure

In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures, i.e., measures of dependence between two multivariate random objects. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall's $τ$, Spearman's $ρ$ and Pearson's correlation coefficient. Among these, only multivariate Kendall's $τ$ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall's $τ$ in this context. To this end, limit theorems for multivariate Kendall's $τ$ are established under the assumption of near-epoch dependent data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall's $τ$ and Pearson's correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall's $τ$ and ordinal pattern dependence.

math.ST↗

arXiv2021

Ordinal Pattern Dependence in the Context of Long-Range Dependence

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

math.ST↗

arXiv2020

Ordinal Patterns in Long-Range Dependent Time Series

We analyze the ordinal structure of long-range dependent time series. To this end, we use so called ordinal patterns which describe the relative position of consecutive data points. We provide two estimators for the probabilities of ordinal patterns and prove limit theorems in different settings, namely stationarity and (less restrictive) stationary increments. In the second setting, we encounter a Rosenblatt distribution in the limit. We prove more general limit theorems for functions with Hermite rank 1 and 2. We derive the limit distribution for an estimation of the Hurst parameter $H$ if it is higher than 3/4. Thus, our theorems complement results for lower values of $H$ which can be found in the literature. Finally, we provide some simulations that illustrate our theoretical results.

math.ST↗

arXiv2020

Ordinal Patterns in Clusters of Subsequent Extremes of Regularly Varying Time Series

In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine.

math.ST↗

arXiv2019

The Fourth Characteristic of a Semimartingale

We extend the class of semimartingales in a natural way. This allows us to incorporate processes having paths that leave the state space R^d. In particular Markov processes related to sub-Markovian kernels, but also non-Markovian processes with path-dependent behavior. By carefully distinguishing between two killing states, we are able to introduce a fourth semimartingale characteristic which generalizes the fourth part of the Lévy quadruple. Using the probabilistic symbol, we analyze the close relationship between the generators of certain Markov processes with killing and their (now four) semimartingale characteristics.

math.PR↗

arXiv2018

Laplace Symbols and Invariant Distributions

We introduce a new kind of symbol in the framework of Itô processes which are bounded on one side. The connection between this symbol and the infinitesimal generator is analyzed. Based on this concept, an integral criterion for invariant distributions of the underlying process is derived. Some applications are mentioned.

math.PR↗

arXiv2016

Criteria for the Finiteness of the Strong $p$-Variation for Lévy-type Processes

Using generalized Blumenthal--Getoor indices, we obtain criteria for the finiteness of the $p$-variation of Lévy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations (SDEs), certain Feller processes and solutions of Lévy driven SDEs. The class of processes is wider than in earlier contributions and using fine continuity we are able to handle general measurable subsets of $R^d$ as state spaces. Furthermore, in contrast to previous contributions on the subject, we introduce a local index in order to complement the upper index. This local index yields a sufficient condition for the infiniteness of the $p$-variation. We discuss various examples in order to demonstrate the applicability of the method.

math.PR↗

arXiv2015

Time Change Equations for Lévy Type Processes

In this paper we analyse time change equations (TCEs) for Lévy-type processes in detail. To this end we establish a connection between TCEs and classical one-dimensional initial value problems (IVPs) which are easier to handle. Properties of the IVPs are linked with properties of the TCEs. We show in a general setting existence and uniqueness of solutions of the TCEs. Our main result is based on the general path properties for Lévy-type processes found in Schnurr (2013). Applications include an existence result for processes which correspond to a certain class of given symbols.

math.PR↗