Lutz Mattner
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Lutz Mattner has a total of 8 co-authors. Here are the top co-authors by number of shared publications:
- Achim Klenke
Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Mainz, Germany
2 shared publications - Irina Gennad'EVna Shevtsova
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia
2 shared publications - Uwe Roesler
Christian-Albrechts-Universität zu Kiel
2 shared publications - Frauke Mattner
Department of Human Medicine, Faculty of Health, Witten/Herdecke University, Alfred-Herrhausen-Strasse 50, Witten, 58455, Germany, 49 221-8907 ext 8677, 49 211-8907-8314
1 shared publication - Gérard Letac
Université Paul Sabatier
1 shared publication - Lena Jonas
Universität Trier, Fachbereich IV – Mathematik, 54286 Trier, Germany
1 shared publication
Latest Publications
- 17 November 2025
- Published by American Mathematical Society (AMS) in Theory of Probability and Mathematical Statistics
- Vol. 113, 113-132
A sharper Lyapunov–Katz central limit error bound for i.i.d. summands Zolotarev-close to normal
We prove a central limit error bound for convolution powers of laws with finite moments of order r ∈ ] 2 , 3 ] r \in ]2,3] , taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of r = 3 r=3 of the sharpening of the Berry (1941) – Esseen (1942) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov’s (1901) theorem. Our proof uses a partial generalisation of the theorem of Senatov and Zolotarev (1986) used for the earlier special case. A result more general than our main one could be obtained by using instead another theorem of Senatov (1980), but, unfortunately, an auxiliary inequality used in the latter’s proof is wrong.
- 17 November 2025
- Published by American Mathematical Society (AMS) in Theory of Probability and Mathematical Statistics
- Vol. 113, 113-132
A sharper Lyapunov–Katz central limit error bound for i.i.d. summands Zolotarev-close to normal
We prove a central limit error bound for convolution powers of laws with finite moments of order r ∈ ] 2 , 3 ] r \in ]2,3] , taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of r = 3 r=3 of the sharpening of the Berry (1941) – Esseen (1942) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov’s (1901) theorem. Our proof uses a partial generalisation of the theorem of Senatov and Zolotarev (1986) used for the earlier special case. A result more general than our main one could be obtained by using instead another theorem of Senatov (1980), but, unfortunately, an auxiliary inequality used in the latter’s proof is wrong.
- 30 October 2024
- Published by American Mathematical Society (AMS) in Theory of Probability and Mathematical Statistics
- Vol. 111, 45-122
A convolution inequality, yielding a sharper Berry–Esseen theorem for summands Zolotarev-close to normal
The classical Berry–Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment with a weak norm distance to normality, using Zolotarev’s ζ \zeta norms. We thus sharpen and simplify two results of Ul’yanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Kantorovich distances (often called Wasserstein distances, the L 1 ^1 norms of differences of distribution functions) of the corresponding factors, and also in the pair of the Lipschitz constants. Side results include a short introduction to ζ \zeta norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and Götze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.
- 30 October 2024
- Published by American Mathematical Society (AMS) in Theory of Probability and Mathematical Statistics
- Vol. 111, 45-122
A convolution inequality, yielding a sharper Berry–Esseen theorem for summands Zolotarev-close to normal
The classical Berry–Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment with a weak norm distance to normality, using Zolotarev’s ζ \zeta norms. We thus sharpen and simplify two results of Ul’yanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Kantorovich distances (often called Wasserstein distances, the L 1 ^1 norms of differences of distribution functions) of the corresponding factors, and also in the pair of the Lipschitz constants. Side results include a short introduction to ζ \zeta norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and Götze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.
- Open Access
A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal
The classical Berry-Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment by a weak norm distance to normality. We thus sharpen and simplify two results of Ulyanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Wasserstein distances (L$^1$ norms of differences of distribution functions) of the corresponding factors, and also, inessentially for the present application, in the pair of the Lipschitz constants. Side results include a short introduction to $\zeta$ norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and G\"otze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.Open AccessA convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal
The classical Berry-Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment by a weak norm distance to normality. We thus sharpen and simplify two results of Ulyanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Wasserstein distances (L$^1$ norms of differences of distribution functions) of the corresponding factors, and also, inessentially for the present application, in the pair of the Lipschitz constants. Side results include a short introduction to $\zeta$ norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and G\"otze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations. - 1 January 2019
- Published by Institute for Applied and Pure Mathematics (IMPA) in Latin American Journal of Probability and Mathematical Statistics
- Vol. 16 (1) , 463-530
An optimal Berry–Esseen type theorem for integrals of smooth functions
- 1 January 2019
- Published by Institute for Applied and Pure Mathematics (IMPA) in Latin American Journal of Probability and Mathematical Statistics
- Vol. 16 (1) , 463-530
An optimal Berry–Esseen type theorem for integrals of smooth functions
- Open Access
An optimal Berry-Esseen type theorem for integrals of smooth functions
We prove a Berry-Esseen type inequality for approximating expectations of sufficiently smooth functions $f$, like $f=|\cdot|^3$, with respect to standardized convolutions of laws $P_1,\ldots, P_n$ on the real line by corresponding expectations based on symmetric two-point laws $Q_1,\ldots,Q_n$ isoscedastic to the $P_i$. Equality is attained for every possible constellation of the Lipschitz constant $\|f"\|^{}_{\mathrm{L}}$ and the variances and the third centred absolute moments of the $P_i$. The error bound is strictly smaller than $\frac 16$ times the Lyapunov ratio times $\|f"\|^{}_{\mathrm{L}}$, and tends to zero also if $n$ is fixed and the third standardized absolute moments of the $P_i$ tend to one. In the homoscedastic case of equal variances of the $P_i$, and hence in particular in the i.i.d. case, the approximating law is a standardized symmetric binomial one. The inequality is strong enough to yield for some constellations, in particular in the i.i.d. case with $n$ large enough given the standardized third absolute moment of $P_1$, an improvement of a more classical and already optimal Berry-Esseen type inequality of Tyurin (2009). Auxiliary results presented include some inequalities either purely analytical or concerning Zolotarev's $\zeta$-metrics, and some binomial moment calculations.Open AccessAn optimal Berry-Esseen type theorem for integrals of smooth functions
We prove a Berry-Esseen type inequality for approximating expectations of sufficiently smooth functions $f$, like $f=|\cdot|^3$, with respect to standardized convolutions of laws $P_1,\ldots, P_n$ on the real line by corresponding expectations based on symmetric two-point laws $Q_1,\ldots,Q_n$ isoscedastic to the $P_i$. Equality is attained for every possible constellation of the Lipschitz constant $\|f"\|^{}_{\mathrm{L}}$ and the variances and the third centred absolute moments of the $P_i$. The error bound is strictly smaller than $\frac 16$ times the Lyapunov ratio times $\|f"\|^{}_{\mathrm{L}}$, and tends to zero also if $n$ is fixed and the third standardized absolute moments of the $P_i$ tend to one. In the homoscedastic case of equal variances of the $P_i$, and hence in particular in the i.i.d. case, the approximating law is a standardized symmetric binomial one. The inequality is strong enough to yield for some constellations, in particular in the i.i.d. case with $n$ large enough given the standardized third absolute moment of $P_1$, an improvement of a more classical and already optimal Berry-Esseen type inequality of Tyurin (2009). Auxiliary results presented include some inequalities either purely analytical or concerning Zolotarev's $\zeta$-metrics, and some binomial moment calculations. - Open Access
The medians for exponential families and the normal law
Let $P$ a probability on the real line generating a natural exponential family $(P_t)_{t\in \R}$. We show that $t$ is a median of $P_t$ for all $t$ only if $P$ is the standard Gaussian law $N(0.1).$ The proof is based on the Choquet Deny equation.Open AccessThe medians for exponential families and the normal law
Let $P$ a probability on the real line generating a natural exponential family $(P_t)_{t\in \R}$. We show that $t$ is a median of $P_t$ for all $t$ only if $P$ is the standard Gaussian law $N(0.1).$ The proof is based on the Choquet Deny equation. - Open Access
Confidence intervals for average success probabilities
We provide Buehler-optimal one-sided and some valid two-sided confidence intervals for the average success probability of a possibly inhomogeneous fixed length Bernoulli chain, based on the number of observed successes. Contrary to some claims in the literature, the one-sided Clopper-Pearson intervals for the homogeneous case are not completely robust here, not even if applied to hypergeometric estimation problems.
Open AccessConfidence intervals for average success probabilities
We provide Buehler-optimal one-sided and some valid two-sided confidence intervals for the average success probability of a possibly inhomogeneous fixed length Bernoulli chain, based on the number of observed successes. Contrary to some claims in the literature, the one-sided Clopper-Pearson intervals for the homogeneous case are not completely robust here, not even if applied to hypergeometric estimation problems.
- Open Access
Confidence bounds for the sensitivity lack of a less specific diagnostic test, without gold standard
We consider the problem of comparing two diagnostic tests based on a sample of paired test results without true state determinations, in cases where the second test can reasonably be assumed to be at least as specific as the first. For such cases, we provide two informative confidence bounds: A lower one for the prevalence times the sensitivity gain of the second test with respect to the first, and an upper one for the sensitivity of the first test. Neither conditional independence of the two tests nor perfectness of any of them needs to be assumd. An application of the proposed confidence bounds to a sample of 256 pairs of laboratory test results for toxigenic Clostridium difficile provides evidence for a dramatic sensitivity gain through first appropriately culturing Clostridium difficile from stool samples before applying an enzyme-immuno-assay.
Open AccessConfidence bounds for the sensitivity lack of a less specific diagnostic test, without gold standard
We consider the problem of comparing two diagnostic tests based on a sample of paired test results without true state determinations, in cases where the second test can reasonably be assumed to be at least as specific as the first. For such cases, we provide two informative confidence bounds: A lower one for the prevalence times the sensitivity gain of the second test with respect to the first, and an upper one for the sensitivity of the first test. Neither conditional independence of the two tests nor perfectness of any of them needs to be assumd. An application of the proposed confidence bounds to a sample of 256 pairs of laboratory test results for toxigenic Clostridium difficile provides evidence for a dramatic sensitivity gain through first appropriately culturing Clostridium difficile from stool samples before applying an enzyme-immuno-assay.
- Open Access
- 1 February 2011
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 139 (02) , 735
Optimal $L_{}^{1}$-bounds for submartingales
The optimal function satisfying <!-- MATH \begin{displaymath} \mathbb{E} |\sum_{1}^n X_i | \ge f(\mathbb{E}|X_1|,\ldots,\mathbb{E}|X_n|) \end{displaymath} -->- 1 February 2011
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 139 (02) , 735
Open AccessOptimal $L_{}^{1}$-bounds for submartingales
The optimal function satisfying <!-- MATH \begin{displaymath} \mathbb{E} |\sum_{1}^n X_i | \ge f(\mathbb{E}|X_1|,\ldots,\mathbb{E}|X_n|) \end{displaymath} --> - 1 June 2010
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 42 (02) , 392-410
Stochastic ordering of classical discrete distributions
For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤st Q can be characterized by their extreme tail ordering equivalent to P({k *})/Q({k *}) ≥ 1 ≥ lim k→k * P({k})/Q({k}), with k * and k * denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k *})/Q({k *}) for finite k *. This includes in particular all pairs where P and Q are both binomial (b n 1,p 1 ≤st b n 2,p 2 if and only if n 1 ≤ n 2 and (1 - p 1) n 1 ≥ (1 - p 2) n 2 , or p 1 = 0), both negative binomial (b − r 1,p 1 ≤st b − r 2,p 2 if and only if p 1 ≥ p 2 and p 1 r 1 ≥ p 2 r 2 ), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).- 1 June 2010
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 42 (02) , 392-410
Stochastic ordering of classical discrete distributions
For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤st Q can be characterized by their extreme tail ordering equivalent to P({k *})/Q({k *}) ≥ 1 ≥ lim k→k * P({k})/Q({k}), with k * and k * denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k *})/Q({k *}) for finite k *. This includes in particular all pairs where P and Q are both binomial (b n 1,p 1 ≤st b n 2,p 2 if and only if n 1 ≤ n 2 and (1 - p 1) n 1 ≥ (1 - p 2) n 2 , or p 1 = 0), both negative binomial (b − r 1,p 1 ≤st b − r 2,p 2 if and only if p 1 ≥ p 2 and p 1 r 1 ≥ p 2 r 2 ), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).