Publications

We extend the theory of Tukey depth to the setting of absolutely regular (\(\beta\)-mixing) stochastic processes. In particular, we generalize Massé's weak convergence result for the Tukey depth process to dependent observations and establish convergence rates expressed in terms of the mixing coefficients. An application extends the theory developed by Massé on depth weighted estimators. First, we show that the assumptions on the weight can be significantly weakened, and secondly we show that the observations can be taken as weakly dependent. A simulation study demonstrates the advantages of depth weighted estimators under heavy tailed innovations.

This article establishes maximal moment inequalities for random variables satisfying absolute regularity, uniformly over possibly infinite \(\mathcal{F}\). Our approach combines admissible partition sequences theory with Pollard's truncation technique and recursive Berbee–Goldstein maximal coupling. We discuss regularity via the Ossiander entropy integral and explain the structural connection with the Boucheron–Bousquet–Lugosi–Massart inequality. Concentration properties are similar to those of Johnson–Schechtman–Zinn and Kwapień–Szulga up to possibly enlarged norms.

This article gives a uniform moment inequality for empirical processes indexed by measurable functions arising from absolutely regular random variables. The bound involves, as is the case in the classical Koltchinskii–Pollard inequality, the uniform entropy integral with an additional factor controlling the minimal measurable cover function. The norm in which this envelope is expressed is related to the dependent Marcinkiewicz–Zygmund inequalities. A detailed overview of minimal requirements to ensure measurability and finiteness of the respective Rosenthal type norms is provided.

We introduce a new class of location-scale models for dynamic functional data in arbitrary but fixed dimensions, where the location and scale functional parameters can evolve over time. A key feature of the parameter dynamics in these models is its observation-driven nature, where the one-step-ahead evolution is fully determined conditional on past observations, yet remains stochastic unconditionally. We estimate the model using a likelihood-based approach designed for sparsely observed data and establish the consistency and asymptotic normality of the underlying static parameters that govern the location-scale dynamics. The choice of objective function and the construction of the dynamics together shield the time-varying location and scale parameters from the potentially distorting effects of influential observations. Simulations reveal that our method can recover the unobserved location-scale dynamics from sparse data, even in the presence of model mis-specification and substantial outliers. We apply our framework to examine the intraday volatility dynamics of Pfizer stock returns during the COVID-19 pandemic, and PM2.5 concentrations measured by low-cost sensors across Europe. The proposed model exhibits robust performance in capturing dynamics for both datasets despite the presence of many large shocks.