Integrated autoregressive conditional duration (ACD) models serve as counterparts to integrated generalized autoregressive conditional heteroskedastic models used for financial returns. However, despite their resemblance, asymptotic theory for ACD is still incomplete. Central challenges arise from the facts that (i) integrated ACD processes imply durations with infinite expectation and (ii) conventional asymptotic approaches break down due to the randomness in the number of durations within a fixed observation period. We fill this gap in the literature and provide a unified asymptotic theory for the (quasi)maximum likelihood estimator for integrated ACD models. Based on the new theoretical 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 exchange traded fund (ETF) trading data. Motivated by parameter estimates near the integrated ACD region, 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 five cryptocurrency ETFs examined, and we reject the integrated ACD hypothesis in favour of heavier-tailed alternatives for four out of the five ETFs.

We consider a borderline case: The central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the i.i.d. case, a well-known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index . In the dependent case, we assume the stronger condition of sequential regular variation of the time series with tail index . We assume that a sample of size from this time series can be split into blocks of size such that as and that the block sums are asymptotically independent. Then we apply classical central limit theory for row-wise i.i.d. 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 -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 .

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.

Based on the GARCH literature, Engle and Russell (1998) established consistency and asymptotic normality of the QMLE for the autoregressive conditional duration (ACD) model, assuming strict stationarity and ergodicity of the durations. Using novel arguments based on renewal process theory, we show that their results hold under the stronger requirement that durations have finite expectation. However, we demonstrate that this is not always the case under the assumption of stationary and ergodic durations. Specifically, we provide a counterexample where the MLE is asymptotically mixed normal and converges at a rate significantly slower than usual. The main difference between ACD and GARCH asymptotics is that the former must account for the number of durations in a given time span being random. As a by-product, we present a new lemma which can be applied to analyze asymptotic properties of extremum estimators when the number of observations is random.

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), 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 behavior 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.

We reconsider 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 (2009), 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, such as 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.

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.

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.

We consider ourselves very fortunate to have been able to publish our 1997 book Embrechts et al. (1997), with the title Modelling Extremal Events for Insurance and Finance, under the expert guidance of Catriona Byrne. Whereas, at the time, Extreme Value Theory (EVT) already had a rich history of methodological developments, our book greatly contributed to new areas of applications. By now, EVT constitutes an important area of research within probability and statistics. What started in 1997 as a book project under the editorial umbrella of Catriona has meanwhile grown into different directions of research for the three authors. We present three very different examples of recent research from the realm of EVT. It gives us great pleasure to contribute to these festive lecture notes, edited in honor of Catriona Byrne.

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 blocks methods to infer features of cluster processes. Our inferential setting promotes 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 ℓα−blocks, where α>0 is the index of regular variation of the series. We focus on inference of important indices in extreme value theory, e.g., the extremal index.

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.

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.

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 $(2+\delta)$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.

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.