arXiv subjects

Georg Köstenberger

Publications and source records attributed to Georg Köstenberger.

5 recordsLinked to original sources

arXiv2026

Quantitative Halmos-Savage theorems and robust large financial markets

We establish a quantitative version of the classical Halmos-Savage Theorem for convex, potentially non-dominated sets of probability measures and its dual counterpart, generalizing previous quantitative versions. These results are then used to derive robust versions of the fundamental theorem of asset pricing (FTAP) in large financial markets in a one-period setting, characterizing the absence of arbitrage under Knightian uncertainty. To this end, we consider robust formulations of no asymptotic arbitrage of first kind (NAA1), which is the large market analogue of ``No unbounded profit with bounded risk'' (NUPBR), as well as no asymptotic arbitrage of second kind (NAA2). Finally, we characterize asymptotic arbitrage of first and second kind in the robust one-period binomial model in terms of the model parameters.

math.PR↗

arXiv2026

Can the $L^1$-$L^\infty$ duality be restored for non-dominated families of probability measures?

The duality $L^{\infty}\simeq (L^{1})'$ frequently breaks down in the presence of model uncertainty, where a single reference measure $P$ is replaced by a non-dominated family of probability measures $\mathcal{P}$. The unavailability of classical measure-theoretic and functional-analytic tools in this regime poses a significant obstacle to developing robust probabilistic frameworks. We show that this duality can be restored for a broad class of robust statistical models by extending the underlying probability space. Specifically, on the extended model, the space $\mathbb{L}^{\infty}(\mathcal{P})$ of $\mathcal{P}$-quasi-surely bounded functions is isometrically isomorphic to the dual of the space of finite signed measures absolutely continuous with respect to at least one element of $\mathcal{P}$. The proposed extension is canonical: it is the smallest $\mathcal{P}$-complete extension of the original $σ$-algebra for which $\mathbb{L}^{\infty}(\mathcal{P})$ is the dual of any normed space. Our assumptions encompass several prominent non-dominated settings, including infinite product measures, Gaussian processes, the Black-Scholes model with uncertain constant volatility and drift, robust binomial models, and, more generally, infinite sequences from any parametric model with almost surely estimable parameters. Furthermore, we unify the existing frameworks of Cohen (2012) and Liebrich et al. (2022), demonstrating that our construction is equivalent to the capacity-based approach under mild assumptions satisfied by the aforementioned examples. Finally, we apply our theory to extend Kraft's (1955) characterization of strictly unbiased hypothesis tests to non-dominated cases.

math.PR↗

arXiv2024

Robust Signal Recovery in Hadamard Spaces

We analyze the stability of (strong) laws of large numbers in Hadamard spaces with respect to distributional perturbations. For the inductive means of a sequence of independent, but not necessarily identically distributed random variables, we provide a concentration inequality in quadratic mean, as well as a strong law of large numbers, generalizing a classical result of K.-T. Sturm. For the Fréchet mean, we generalize H. Ziezold's law of large numbers in Hadamard spaces. In this case, we neither require our data to be independent, nor identically distributed; reasonably mild conditions on the first two moments of our sample are enough. Additionally, we look at data contamination via a model inspired by Huber's $\varepsilon$-contamination model, in which we replace a random portion of the data with noise. In the most general setup, we do neither require the data, nor the noise to be i.i.d., nor do we require the noise to be independent of the data. To analyze the stability of the (non-symmetric) inductive mean with respect to data loss, data permutation, and noise, a resampling scheme is introduced, and sufficient conditions for its convergence are provided. These results suggest that means in Hadamard spaces are as robust as in Euclidean spaces. This is underlined by a small simulation study, in which we compare the robustness of means on the manifold of positive definite matrices, with means on open books.

math.ST↗

arXiv2024

Sharp oracle inequalities and universality of the AIC and FPE

In two landmark papers, Akaike introduced the AIC and FPE, demonstrating their significant usefulness for prediction. In subsequent seminal works, Shibata developed a notion of asymptotic efficiency and showed that both AIC and FPE are optimal, setting the stage for decades-long developments and research in this area and beyond. Conceptually, the theory of efficiency is universal in the sense that it (formally) only relies on second-order properties of the underlying process $(X_t)_{t\in \mathbb{Z}}$, but, so far, almost all (efficiency) results require the much stronger assumption of a linear process with independent innovations. In this work, we establish sharp oracle inequalities subject only to a very general notion of weak dependence, establishing a universal property of the AIC and FPE. A direct corollary of our inequalities is asymptotic efficiency of these criteria. Our framework contains many prominent dynamical systems such as random walks on the regular group, functionals of iterated random systems, functionals of (augmented) Garch models of any order, functionals of (Banach space valued) linear processes, possibly infinite memory Markov chains, dynamical systems arising from SDEs, and many more.

math.ST↗

arXiv2021

Weingarten Calculus

We consider the problem of computing the integral $$ \int_{\mathcal{U}(d)} u_{i_1j_1}\cdots u_{i_nj_n} \bar{u}_{i'_1j'_1} \cdots \bar{u}_{i'_{n'}j'_{n'}} dU, $$ where the integration takes place with respect to the probability Haar measure on the unitary group $\mathcal{U}(d)$, and the $u_{ij}$ denotes the $ij$-th entry of a unitary matrix $U$. We present a unified approach connecting classical results, the explicit formula for the integral given by B. Collins and P. Sniady and subsequent works of various authors providing different points of view. Finally we are able to provide an explicit formula for the $2n$-th moment of the trace of a unitary Haar random matrix, generalizing a result of P. Diaconis.

math.PR↗