arXiv subjects

Christian Döbler

Publications and source records attributed to Christian Döbler.

At least 19 recordsLinked to original sources

arXiv2026

On the Malliavin calculus on product spaces and an infinite de Jong theorem

We extend the Malliavin theory for $L^2$-functionals on product probability spaces that has recently been developed independently by Decreusefond and Halconruy (2019) and by Duerinckx (2021), by characterizing the domains and investigating the actions of the three Malliavin operators in terms of the infinite Hoeffding decomposition in $L^2$, which we identify as the natural analogue of the famous Wiener-Itô chaos decomposition on Gaussian and Poisson spaces. We further explore the corresponding Ornstein-Uhlenbeck semigroup in terms of its Mehler representation and prove new moment bounds for iterated gradients. As an illustration of the abstract framework, we prove an infinite version of the quantitative de Jong CLT that has recently been proved by G. Peccati and the author (2017) and by the author (2024).

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arXiv2025

On existence and uniqueness of univariate Stein kernels

We completely characterize the class of univariate distributions allowing for a Stein kernel and illustrate our result by means of some concrete distributions. Moreover, we apply our findings to prove a quantitative version of the central limit theorem with optimal rate $n^{-1/2}$ in total variation distance for i.i.d. random variables whose distribution belongs to that class.

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arXiv2025

Discrete probability spaces revisited

We give an elementary proof of the known fact that every probability measure, defined on an arbitrary $σ$-field on a countable sample space $Ω$, may in fact be extended to a probability measure on the power set of $Ω$. This result is further discussed and motivated in the context of discrete random variables.

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arXiv2024

New bounds for normal approximation on product spaces with applications to monochromatic edges, random sums and an infinite de Jong CLT

We extend the Malliavin theory for $L^2$-functionals on product probability spaces that has recently been developed by Decreusefond and Halconruy (2019) and by Duerinckx (2021), by characterizing the domains and investigating the actions of the three Malliavin operators in terms of the infinite Hoeffding decomposition in $L^2$, which we identify as the natural analogue of the famous Wiener-Itô chaos decomposition on Gaussian and Poisson spaces. We further combine this theory with Stein's method for normal approximation in order to provide three different types of abstract Berry-Esseen and Wasserstein bounds: a) Malliavin-Stein bounds involving the Malliavin gradient $D$ and the pseudo-inverse of the Ornstein-Uhlenbeck generator $L$, b) bounds featuring the carré-du-champ operator $Γ$ and c) bounds making use of a Clark-Ocone type integration-by-parts formula. To demonstrate the flexibility of these abstract bounds, we derive quantitative central limit theorems for the number of monochromatic edges in a uniform random coloring of a graph sequence as well as for random sums and prove an infinite version of the quantitative de Jong CLT that has recently been proved by G. Peccati and the author (2017) and by the author (2023). As a further theoretical application, we deduce new abstract Berry-Esseen and Wasserstein bounds for functionals of a general independent Rademacher sequence.

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arXiv2023

The Berry-Esseen bound in de Jong's CLT

We prove a Berry-Esseen bound in de Jong's classical CLT for normalized, completely degenerate $U$-statistics, which says that the convergence of the fourth moment sequence to three and a Lindeberg-Feller type negligibility condition are sufficient for asymptotic normality. Our bound is of the same optimal order as the bound on the Wasserstein distance to normality that has recently been proved by Döbler and Peccati (2017).

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arXiv2022

The multivariate functional de Jong CLT

We prove a multivariate functional version of de Jong's CLT (1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on $[0,1]$ weakly converge in the Skorohod space as soon as their fourth cumulants in $t=1$ vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the `universality of Wiener chaos' phenomenon first observed in Nourdin, Peccati and Reinert (2010).

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arXiv2022

Functional Convergence of Sequential U-processes with Size-Dependent Kernels

We consider sequences of $U$-processes based on symmetric kernels of a fixed order, that possibly depend on the sample size. Our main contribution is the derivation of a set of analytic sufficient conditions, under which the aforementioned $U$-processes weakly converge to a linear combination of time-changed independent Brownian motions. In view of the underlying symmetric structure, the involved time-changes and weights remarkably depend only on the order of the U-statistic, and have consequently a universal nature. Checking these sufficient conditions requires calculations that have roughly the same complexity as those involved in the computation of fourth moments and cumulants. As such, when applied to the degenerate case, our findings are infinite-dimensional extensions of the central limit theorems (CLTs) proved in de Jong (1990) and Döbler and Peccati (2017). As important tools in our analysis, we exploit the multidimensional central limit theorems established in Döbler and Peccati (2019) together with upper bounds on absolute moments of degenerate $U$-statistics by Ibragimov and Sharakhmetov (2002), and also prove some novel multiplication formulae for degenerate symmetric $U$-statistics -- allowing for different sample sizes -- that are of independent interest. We provide applications to random geometric graphs and to a class of $U$-statistics of order two, whose Gaussian fluctuations have been recently studied by Robins et al. (2016), in connection with quadratic estimators in non-parametric models. In particular, our application to random graphs yields a class of new functional central limit theorems for subgraph counting statistics, extending previous findings in the literature. Finally, some connections with invariance principles in changepoint analysis are established.

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arXiv2021

A short proof of Lévy's continuity theorem without using tightness

In this note we present a new short and direct proof of Lévy's continuity theorem in arbitrary dimension $d$, which does not rely on Prohorov's theorem, Helly's selection theorem or the uniqueness theorem for characteristic functions. Instead, it is based on convolution with a small (scalar) Gaussian distribution as well as on basic facts about weak convergence and measure theory. Moreover, we show how, by similar means, one may prove the fact that a distribution with integrable characteristic function is absolutely continuous with respect to $d$-dimensional Lebesgue measure and derive the formula for its density.

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arXiv2021

The fourth moment theorem on the Poisson space

We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for several years -- shows that the so-called `fourth moment phenomenon', first discovered by Nualart and Peccati (2005) in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a non-diffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux (2012) and Azmoodeh, Campese and Poly (2014). To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of non-linear functionals of a Poisson measure.

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arXiv2021

Quantitative CLTs for symmetric $U$-statistics using contractions

We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of {\it contraction operators}. Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called `dense parameter regime' for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).

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arXiv2021

Stein's method of exchangeable pairs in multivariate functional approximations

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying it to joint subgraph counts in an Erdős-Renyi random graph model on the one hand and to vectors of weighted, degenerate $U$-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.

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arXiv2020

Normal approximation via non-linear exchangeable pairs

We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.

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arXiv2018

Recurrence and Transience of Frogs with Drift on $\mathbb{Z}^d$

We study the frog model on $\mathbb{Z}^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

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arXiv2018

Fourth moment theorems on the Poisson space: analytic statements via product formulae

We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carré-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Döbler and Peccati (2018) and Döbler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

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arXiv2018

Fourth moment theorems on the Poisson space in any dimension

We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. Döbler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case. Finally, a transfer principle "from-Poisson-to-Gaussian" is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.

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arXiv2017

An iterative technique for bounding derivatives of solutions of Stein equations

We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations $Lf=h-\mathbb{E}h(Z)$, where $L$ is a linear differential operator and $Z$ is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function $h$. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive bounds for derivatives of any order of the solution of the general variance-gamma Stein equation.

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arXiv2017

On the fourth moment condition for Rademacher chaos

Adapting the spectral viewpoint suggested in Ledoux (2012) in the context of symmetric Markov diffusion generators and recently exploited in the non-diffusive setup of a Poisson random measure by Döbler and Peccati (2017), we investigate the fourth moment condition for discrete multiple integrals with respect to general, i.e.\ non-symmetric and non-homogeneous, Rademacher sequences and show that, in this situation, the fourth moment alone does not govern the asymptotic normality. Indeed, here one also has to take into consideration the maximal influence of the corresponding kernel functions. In particular, we show that there is no exact fourth moment theorem for discrete multiple integrals of order $m\geq2$ with respect to a symmetric Rademacher sequence. This behavior, which is in contrast to the Gaussian (see Nualart and Peccati (2005)) and Poisson (see Döbler and Peccati (2017)) situation, closely resembles the conditions for asymptotic normality of degenerate, non-symmetric $U$-statistics from the classical paper by de Jong (1990).

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arXiv2017

The Gamma Stein equation and non-central de Jong theorems

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate U-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (1990), Nourdin, Peccati and Reinert (2010) and Döbler and Peccati (2016), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thäle (2013), and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (2009). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

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