arXiv subjects

Thomas Mikosch

Publications and source records attributed to Thomas Mikosch.

At least 19 recordsLinked to original sources

arXiv2026

Moments for self-normalized partial sums

We consider a regularly varying stationary sequence of random variables (Xt) with tail index ___ < 2. For these sequences we study the joint convergence of sums, `p- type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes (Xt) whose limit ratios possess all moments. But, in contrast to the latter situation, there also exist sequences (Xt) where certain moments of the limit ratio are in___nite. This phenomenon results from extremal clusters in the sequence.

math.PR↗

arXiv2026

Bootstrap inference in autoregressive duration models

This paper develops bootstrap inference for autoregressive conditional duration (ACD) models observed over a fixed calendar span, so that the number of durations is random. We study recursive schemes that either fix the calendar span or the realized event count. For the fixed-count bootstrap, we establish consistency when the duration tail index satisfies $\kappa\geq1$. When $0<\kappa<1$, classical consistency fails because the estimator has a mixed-normal limit, but the bootstrap reproduces its conditional Gaussian component. Consequently, basic percentile intervals remain first-order valid and bootstrap $t$-statistics are asymptotically standard normal. Monte Carlo experiments show accurate finite-sample inference across finite- and infinite-mean regimes and robustness to non-exponential innovations. An application to cryptocurrency ETF transaction durations finds strong persistence and illustrates the practical difference between fixed-count and random-count inference.

econ.EM↗

arXiv2025

Beyond the Mean: Limit Theory and Tests for Infinite-Mean Autoregressive Conditional Durations

Integrated autoregressive conditional duration (ACD) models serve as natural counterparts to the well-known integrated GARCH models used for financial returns. However, despite their resemblance, asymptotic theory for ACD is challenging and also not complete, in particular for integrated ACD. Central challenges arise from the facts that (i) integrated ACD processes imply durations with infinite expectation, and (ii) even in the non-integrated case, conventional asymptotic approaches break down due to the randomness in the number of durations within a fixed observation period. Addressing these challenges, we provide here unified asymptotic theory for the (quasi-) maximum likelihood estimator for ACD models; a unified theory which includes integrated ACD models. Based on the new results, we also provide a novel framework for hypothesis testing in duration models, enabling inference on a key empirical question: whether durations possess a finite or infinite expectation. We apply our results to high-frequency cryptocurrency ETF trading data. Motivated by parameter estimates near the integrated ACD boundary, we assess whether durations between trades in these markets have finite expectation, an assumption often made implicitly in the literature on point process models. Our empirical findings indicate infinite-mean durations for all the five cryptocurrencies examined, with the integrated ACD hypothesis rejected -- against alternatives with tail index less than one -- for four out of the five cryptocurrencies considered.

econ.EM↗

arXiv2025

The Gaussian central limit theorem for a stationary time series with infinite variance

We consider a borderline case: the central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the iid case a well-known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index $\alpha=2$. In the dependent case we assume the stronger condition of sequential regular variation of the time series with tail index $\alpha=2$. We assume that a sample of size $n$ from this time series can be split into $k_n$ blocks of size $r_n\to\infty$ such that $r_n/n\to 0$ as $n\to\infty$ and that the block sums are asymptotically independent. Then we apply classical central limit theory for row-wise iid triangular arrays. The necessary and sufficient conditions for such independent block sums will be verified by using large deviation results for the time series. We derive the central limit theorem for $m$-dependent sequences, linear processes, stochastic volatility processes and solutions to affine stochastic recurrence equations whose marginal distributions have infinite variance and are regularly varying with tail index $\alpha=2$.

math.PR↗

arXiv2023

Asymptotics for the Generalized Autoregressive Conditional Duration Model

Engle and Russell (1998, Econometrica, 66:1127--1162) apply results from the GARCH literature to prove consistency and asymptotic normality of the (exponential) QMLE for the generalized autoregressive conditional duration (ACD) model, the so-called ACD(1,1), under the assumption of strict stationarity and ergodicity. The GARCH results, however, do not account for the fact that the number of durations over a given observation period is random. Thus, in contrast with Engle and Russell (1998), we show that strict stationarity and ergodicity alone are not sufficient for consistency and asymptotic normality, and provide additional sufficient conditions to account for the random number of durations. In particular, we argue that the durations need to satisfy the stronger requirement that they have finite mean.

econ.EM↗

arXiv2023

Self-normalized partial sums of heavy-tailed time series

We study the joint limit behavior of sums, maxima and $\ell^p$-type moduli for samples taken from an $\mathbb{R}^d$-valued regularly varying stationary sequence with infinite variance. As a consequence, we can determine the distributional limits for ratios of sums and maxima, studentized sums, and other self-normalized quantities in terms of hybrid characteristic functions and Laplace transforms. These transforms enable one to calculate moments of the limits and to characterize the differences between the iid and stationary cases in terms of indices which describe effects of extremal clustering on functionals acting on the dependent sequence.

math.PR↗

arXiv2022

The Econometrics of Financial Duration Modeling

We establish new results for estimation and inference in financial durations models, where events are observed over a given time span, such as a trading day, or a week. For the classical autoregressive conditional duration (ACD) models by Engle and Russell (1998, Econometrica 66, 1127-1162), we show that the large sample behavior of likelihood estimators is highly sensitive to the tail behavior of the financial durations. In particular, even under stationarity, asymptotic normality breaks down for tail indices smaller than one or, equivalently, when the clustering behaviour of the observed events is such that the unconditional distribution of the durations has no finite mean. Instead, we find that estimators are mixed Gaussian and have non-standard rates of convergence. The results are based on exploiting the crucial fact that for duration data the number of observations within any given time span is random. Our results apply to general econometric models where the number of observed events is random.

econ.EM↗

arXiv2022

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random field and regularly varying max-moving averages.

math.ST↗

arXiv2022

Large deviations of lp-blocks of regularly varying time series and applications to cluster inference

In the regularly varying time series setting, a cluster of exceedances is a short period for which the supremum norm exceeds a high threshold. We propose to study a generalization of this notion considering short periods, or blocks, with lp-norm above a high threshold. Our main result derives new large deviation principles of extremal lp-blocks, which guide us to define and characterize spectral cluster processes in lp. We then study cluster inference in lp to motivate our results. We design consistent disjoint block estimators to infer features of cluster processes. Our estimators promote the use of large empirical quantiles from the lp-norm of blocks as threshold levels which eases implementation and also facilitates comparison for different p>0. Our approach highlights the advantages of cluster inference based on extremal l$α$--blocks, where $α$>0 is the index of regular variation of the series. We focus on inferring important indices in extreme value theory, e.g., the extremal index.

math.PR↗

arXiv2022

Distance covariance for random fields

We study an independence test based on distance correlation for random fields $(X,Y)$. We consider the situations when $(X,Y)$ is observed on a lattice with equidistant grid sizes and when $(X,Y)$ is observed at random locations. We provide asymptotic theory for the sample distance correlation in both situations and show bootstrap consistency. The latter fact allows one to build a test for independence of $X$ and $Y$ based on the considered discretizations of these fields. We illustrate the performance of the bootstrap test in a simulation study involving fractional Brownian and infinite variance stable fields. The independence test is applied to Japanese meteorological data, which are observed over the entire area of Japan.

math.ST↗

arXiv2021

Some variations on the extremal index

We re-consider Leadbetter's extremal index for stationary sequences. It has interpretation as reciprocal of the expected size of an extremal cluster above high thresholds. We focus on heavy-tailed time series, in particular on regularly varying stationary sequences, and discuss recent research in extreme value theory for these models. A regularly varying time series has multivariate regularly varying finite-dimensional distributions. Thanks to results by Basrak and Segers we have explicit representations of the limiting cluster structure of extremes, leading to explicit expressions of the limiting point process of exceedances and the extremal index as a summary measure of extremal clustering. The extremal index appears in various situations which do not seem to be directly related, like the convergence of maxima and point processes. We consider different representations of the extremal index which arise from the considered context. We discuss the theory and apply it to a regularly varying AR(1) process and the solution to an affine stochastic recurrence equation

math.PR↗

arXiv2020

Gumbel and Fréchet convergence of the maxima of independent random walks

We consider point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. The proofs heavily depend on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.

math.PR↗

arXiv2020

Point process convergence for the off-diagonal entries of sample covariance matrices

We study point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a $(2+δ)$th moment. We make heavily use of precise large deviation results for sums of iid random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.

math.PR↗

arXiv2020

Precise large deviations for dependent subexponential variables

In this paper we study precise large deviations for the partial sums of a stationary sequence with a subexponential marginal distribution. Our main focus is on distributions which either have a regularly varying or a lognormal-type tail. We apply the results to prove limit theory for the maxima of the entries large sample covariance matrices.

math.PR↗

arXiv2020

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrtγ)^2$ and $(1-\sqrtγ)^2$, respectively, as $n \to \infty$, if $p/n\to γ\in (0,1]$ and the truncated variance of the entry distribution is 'almost slowly varying', a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marchenko-Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study. In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.

math.PR↗

arXiv2020

Large sample autocovariance matrices of linear processes with heavy tails

We provide asymptotic theory for certain functions of the sample autocovariance matrices of a high-dimensional time series with infinite fourth moment. The time series exhibits linear dependence across the coordinates and through time. Assuming that the dimension increases with the sample size, we provide theory for the eigenvectors of the sample autocovariance matrices and find explicit approximations of a simple structure, whose finite sample quality is illustrated for simulated data. We also obtain the limits of the normalized eigenvalues of functions of the sample autocovariance matrices in terms of cluster Poisson point processes. In turn, we derive the distributional limits of the largest eigenvalues and functionals acting on them. In our proofs, we use large deviation techniques for heavy-tailed processes, point process techniques motivated by extreme value theory, and related continuous mapping arguments.

math.ST↗

arXiv2020

The eigenstructure of the sample covariance matrices of high-dimensional stochastic volatility models with heavy tails

We consider a $p$-dimensional time series where the dimension $p$ increases with the sample size $n$. The resulting data matrix $X$ follows a stochastic volatility model: each entry consists of a positive random volatility term multiplied by an independent noise term. The volatility multipliers introduce dependence in each row and across the rows. We study the asymptotic behavior of the eigenvalues and eigenvectors of the sample covariance matrix $XX'$ under a regular variation assumption on the noise. In particular, we prove Poisson convergence for the point process of the centered and normalized eigenvalues and derive limit theory for functionals acting on them, such as the trace. We prove related results for stochastic volatility models with additional linear dependence structure and for stochastic volatility models where the time-varying volatility terms are extinguished with high probability when $n$ increases. We provide explicit approximations of the eigenvectors which are of a strikingly simple structure. The main tools for proving these results are large deviation theorems for heavy-tailed time series, advocating a unified approach to the study of the eigenstructure of heavy-tailed random matrices.

math.PR↗

arXiv2019

Homogeneous mappings of regularly varying vectors

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation property of product-type functions of regularly varying random vectors, generalizing and extending the univariate theory in various directions. The main result is then applied to characterize the regular variation property of products of iid regularly varying quadratic random matrices and of solutions to affine stochastic recurrence equations under non-standard conditions.

math.PR↗