Uwe Roesler
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
Research Subjects
Uwe Roesler has a total of 12 co-authors. Here are the top co-authors by number of shared publications:
- Gerold Alsmeyer
University of Münster Technical University of Dresden
3 shared publications Alex Iksanov
2 shared publications- Lutz Mattner
Universität Trier, Fachbereich IV – Mathematik, 54286 Trier, Germany
2 shared publications - Mehri Javanian
Department of Statistics, University of Zanjan, Zanjan, Iran
2 shared publications - Alexander Iksanov
Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
1 shared publication - Ludger Rüschendorf
Department of Mathematical Stochastics, University of Freiburg, Freiburg, Germany
1 shared publication
Latest Publications
- 26 December 2025
- Published by Springer Nature in Journal of Fixed Point Theory and Applications
- Vol. 28 (1) , 7
On the (k, r)-Quicksort on the fly process
- 26 December 2025
- Published by Springer Nature in Journal of Fixed Point Theory and Applications
- Vol. 28 (1) , 7
On the (k, r)-Quicksort on the fly process
- Open Access
On the $(k,r)$-Quicksort on the fly process
, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
Open AccessOn the $(k,r)$-Quicksort on the fly process
, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
- 4 March 2019
- Published by Taylor & Francis in Stochastic Models
- Vol. 35 (3) , 238-251
Stochastic fixed-point equations
The weighted branching process is a natural extension of branching processes. Branching processes count only individuals, weighted branching processes give every individual an abstract weight. Mathematically, we face a (random) dynamical system indexed by a tree. We give an overview of research into this directions, including and unifying many examples of this structure, like Biggins branching random walk and Mandelbrot cascades. WBP are closely connected to stochastic fixed-point equations, fractals and have many applications in genetics, computer science and algorithms. The most advanced example we present is the Quicksort process.
- 4 March 2019
- Published by Taylor & Francis in Stochastic Models
- Vol. 35 (3) , 238-251
Stochastic fixed-point equations
The weighted branching process is a natural extension of branching processes. Branching processes count only individuals, weighted branching processes give every individual an abstract weight. Mathematically, we face a (random) dynamical system indexed by a tree. We give an overview of research into this directions, including and unifying many examples of this structure, like Biggins branching random walk and Mandelbrot cascades. WBP are closely connected to stochastic fixed-point equations, fractals and have many applications in genetics, computer science and algorithms. The most advanced example we present is the Quicksort process.
- 1 December 2018
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 50 (A) , 131-140
All solutions of the stochastic fixed point equation of the Quicksort process
The Quicksort process R (Rösler (2018)) can be characterized as the unique endogenous solution of the inhomogeneous stochastic fixed point equation R=D(UR1(1∧t∕U)+𝟭{U<t}(1-U)R2((t-U)∕(1-U))+C(U,t))t on the space 𝒟 of càdlàg functions, such that R(1) has the Quicksort distribution. In this paper we characterize all 𝒟-valued solutions of that equation. Every solution can be represented as the convolution of a solution of the inhomogeneous equation and a general solution of the homogeneous equation (Rüschendorf (2006)). The general solutions of the homogeneous equation are the distributions of Cauchy processes Y with constant drift. Any distribution of R+Y for independent R and Y is a solution of the inhomogeneous equation. Every solution of the inhomogeneous equation is of the form R+Y, where R and Y are independent. The endogenous solutions for the inhomogeneous equation are the shifted Quicksort process distributions. In comparison, the Quicksort distribution is the endogenous solution of the Quicksort fixed point equation unique up to a constant (Rösler (1991)). The general solution can be represented as the convolution of the shifted Quicksort distribution and some symmetric Cauchy distribution (Fill and Janson (2000)), possibly degenerate.
- 1 December 2018
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 50 (A) , 131-140
All solutions of the stochastic fixed point equation of the Quicksort process
The Quicksort process R (Rösler (2018)) can be characterized as the unique endogenous solution of the inhomogeneous stochastic fixed point equation R=D(UR1(1∧t∕U)+𝟭{U<t}(1-U)R2((t-U)∕(1-U))+C(U,t))t on the space 𝒟 of càdlàg functions, such that R(1) has the Quicksort distribution. In this paper we characterize all 𝒟-valued solutions of that equation. Every solution can be represented as the convolution of a solution of the inhomogeneous equation and a general solution of the homogeneous equation (Rüschendorf (2006)). The general solutions of the homogeneous equation are the distributions of Cauchy processes Y with constant drift. Any distribution of R+Y for independent R and Y is a solution of the inhomogeneous equation. Every solution of the inhomogeneous equation is of the form R+Y, where R and Y are independent. The endogenous solutions for the inhomogeneous equation are the shifted Quicksort process distributions. In comparison, the Quicksort distribution is the endogenous solution of the Quicksort fixed point equation unique up to a constant (Rösler (1991)). The general solution can be represented as the convolution of the shifted Quicksort distribution and some symmetric Cauchy distribution (Fill and Janson (2000)), possibly degenerate.
- 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} --> - Open Access
Optimal L$^1$-bounds for submartingales
The optimal function $f$ satisfying $$ \mathbb{E} |\sum_{1}^n X_i | \ge f(\mathrbb{E}|X_1|,...,\mathbb{E}|X_n|) $$ for every martingale $(X_1,X_1+X_2, ...,\sum_{i=1}^n X_i)$ is shown to be given by $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n \cup \Big\{\frac {a_k}2\Big\}_{k=3}^n $$ for $a\in{[0,\infty[}^n_{}$. A similar result is obtained for submartingales $(0,X_1,X_1+X_2,..., \sum_{i=1}^n X_i)$. The optimality proofs use a convex-analytic comparison lemma of independent interest.Open AccessOptimal L$^1$-bounds for submartingales
The optimal function $f$ satisfying $$ \mathbb{E} |\sum_{1}^n X_i | \ge f(\mathrbb{E}|X_1|,...,\mathbb{E}|X_n|) $$ for every martingale $(X_1,X_1+X_2, ...,\sum_{i=1}^n X_i)$ is shown to be given by $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n \cup \Big\{\frac {a_k}2\Big\}_{k=3}^n $$ for $a\in{[0,\infty[}^n_{}$. A similar result is obtained for submartingales $(0,X_1,X_1+X_2,..., \sum_{i=1}^n X_i)$. The optimality proofs use a convex-analytic comparison lemma of independent interest. - Open Access
Exponential rate of L_p-convergence of intrinsic martingales in supercritical branching random walks
Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero.Open AccessExponential rate of L_p-convergence of intrinsic martingales in supercritical branching random walks
Let $W_n, n\in\mn_{0}$ be an intrinsic martingale with almost sure limit $W$ in a supercritical branching random walk. We provide criteria for the $L_p$-convergence of the series $\sum_{n\ge 0} e^{an}(W-W_n)$ for $p>1$ and $a>0$. The result may be viewed as a statement about the exponential rate of convergence of $\me |W-W_n|^p$ to zero. - 9 April 2008
- Published by Springer Nature in Journal of Theoretical Probability
- Vol. 22 (3) , 666-682
On Distributional Properties of Perpetuities
- 9 April 2008
- Published by Springer Nature in Journal of Theoretical Probability
- Vol. 22 (3) , 666-682
On Distributional Properties of Perpetuities
- Open Access
On distributional properties of perpetuities
We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.Open AccessOn distributional properties of perpetuities
We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.