Poster session 1

Monday, 7/14, 12:00-13:00, Congress Centre

P2: Pricing options on the cryptocurrency futures contracts

Julia Kończal (Wrocław University of Science and Technology)

stand: S2

The cryptocurrency options market is notable for its high volatility and lower liquidity compared to traditional markets. These characteristics introduce significant challenges to traditional option pricing methodologies. Addressing these complexities requires advanced models that can effectively capture the dynamics of the market. We explore which option pricing models are most effective in valuing cryptocurrency options. Specifically, we calibrate and evaluate the performance of the Black—Scholes, Merton Jump Diffusion, Variance Gamma, Kou, Heston, and Bates models. Our analysis focuses on pricing vanilla options on futures contracts for Bitcoin (BTC) and Ether (ETH). We find that the Black–Scholes model exhibits the highest pricing errors. In contrast, the Kou and Bates models achieve the lowest errors, with the Kou model performing the best for the BTC options and the Bates model for ETH options. The results highlight the importance of incorporating jumps and stochastic volatility into pricing models to better reflect the behavior of these assets.

P3: Fractional Brownian motions with random Hurst exponent

Hubert Woszczek (Wrocław University of Science and Technology)

stand: S3

We present several results concerning two types of fractional Brownian motions with a random Hurst exponent, which are inspired by recent biological experiments in single particle tracking. In both cases, we introduce basic probabilistic properties such as the q-th moment of the absolute value of the process, the autocovariance function, and the expectation of the time-averaged mean squared displacement. Furthermore, we analyze the ergodic properties of both processes. Alongside the theoretical analysis, we also provide a numerical study of the results. The poster is based on the results of \([1]\).

Bibliography

\([1]\) Hubert Woszczek, Agnieszka Wyłomańska, Aleksei Chechkin. "Riemann–Liouville Fractional Brownian Motion with Random Hurst Exponent." Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 35, no. 2, 2025, pp. 0243975.

P4: Solution to Stochastic Loewner Equation with Several Complex Variables using Nevanlinna Theory

Andrej Srakar (University of Ljubljana)

stand: S4

Becker (1973) studied solutions to the Loewner differential equation in one complex variable using Carathéodory class of holomorphic functions. However, in several complex variables point singularities are removable and other approaches necessary. Pfaltzgraff generalized to higher dimensions the Loewner differential equation and developed existence and uniqueness theorems for its solutions. The existence and regularity theory has been considered by several authors, and applications given to the characterization of subclasses of biholomorphic mappings, univalence criteria, growth theorems and coefficient bounds for restricted classes of biholomorphic mappings. Duren et al. (2010) studied general form of solutions to the Loewner differential equation under common assumptions of holomorphicity and uniquely determined univalent subordination chains. To our knowledge, to date stochastic Loewner equation has not been studied in a several complex variable setting. We solve the equation in its ordinary and partial differential form by firstly appropriately defining Brownian motion in high dimensions, following Pitman and Yor (2018). We translate the problem in meromorphic form using generalizations of Nevanlinna theory for several complex variables (Noguchi and Winkelman, 2013) and previous work on invariance in the connections of Nevanlinna theory and stochastic processes (e.g. Atsuji, 1995). Finally, the equation is solved using general techniques from rough paths theory (see e.g. Hairer, 2013) and complex Feynman-Kac theorem (see e.g. Grothaus et al., 2010; Xu, 2015). Our solution allows to study stochastic phenomena such as Schramm-Loewner evolution in high dimensions (d>2) which answers one of important present open issues in probability theory and random geometry in particular. We shortly discuss convergence to a scaling limit for multidimensional lattice models as a power series problem on Hartogs domain. Connection between Nevanlinna theory in several complex variables and high-dimensional Brownian motions promises to provide a path to solving stochastic differential equations with several complex variables, which is at present an unaddressed issue in probability theory.

P5: Marcinkiewicz-Zygmund type strong law of large numbers for supOU processes

Danijel Grahovac (University of Osijek, Croatia)

stand: S5

Superpositions of Ornstein-Uhlenbeck type processes (supOU) form a versatile class of infinitely divisible stationary stochastic processes. These processes allow for independent modeling of the marginal distribution and the dependence structure. We present results on the almost sure growth of integrated supOU processes, establishing a Marcinkiewicz-Zygmund type strong law of large numbers. Unlike the classical version, the critical moment condition in our results depends not only on the finiteness of moments but also on the strength of dependence.

We further explore connections to weak limit theorems and demonstrate that the derived growth rates are optimal. Integrated supOU processes may exhibit intermittency, characterized by higher-order moments growing faster than predicted by limit theorems. However, our results demonstrate that large peaks, despite the unusual growth of moments, do not occur infinitely often. We also discuss possible extensions to mixed moving average processes.

The results presented are part of joint work with Péter Kevei (University of Szeged, Hungary), Nikolai N. Leonenko (Cardiff University, UK) and Murad S. Taqqu (Boston University, USA).

Bibliography

\([1]\) D. Grahovac, N.N. Leonenko, A. Sikorskii, M. Taqqu, The unusual properties of aggregated superpositions of Ornstein-Uhlenbeck type processes, Bernoulli 25/3 (2019), 2029-2050

\([2]\) D. Grahovac, N.N. Leonenko, M. Taqqu, Limit theorems, scaling of moments and intermittency for integrated finite variance supOU processes, Stochastic Processes and their Applications 129/12 (2019), 5113-5150

\([3]\) D. Grahovac, N.N. Leonenko, M. Taqqu, The multifaceted behavior of integrated supOU processes: The infinite variance case, Journal of Theoretical Probability 33 (2020), 1801-1831

\([4]\) D. Grahovac, N.N. Leonenko, M. Taqqu, Intermittency and Multiscaling in Limit Theorems, Fractals 30/7 (2022), 1-18

\([5]\) D. Grahovac, P. Kevei, Almost sure growth of integrated supOU processes, Bernoulli (2025), accepted

P6: Time Scale Transformation in Bivariate Pearson Diffusions: A Shift from Light to Heavy Tails

Nenad Šuvak (School of Applied Mathematics and Informatics, University of Osijek, Croatia)

stand: S6

In applications that require addressing heavy-tailed behavior, heavy-tailed Pearson diffusions provide a natural alternative to well-known Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. All three heavy-tailed Pearson diffusions, having inverse gamma, Fisher-Snedecor (\(F\)) and Student stationary distributions, can be constructed via an absolutely continuous time-change employed in a specific functional transformation of CIR or OU process. Moreover, time-change rates in stochastic clocks are continuous functionals of the CIR process. \(F\)-diffusion, heavy-tailed Pearson diffusion with a stationary \(F\) distribution, is derived from the bivariate CIR process \(\left( X_{1}(t), X_{2}(t) \right)_{t \geq 0}\) given by SDE \[\tag{1} \left\{ \begin{array}{l} dX_{1}(t) = -b \left( X_{1}(t) - \beta_{1} \right) \, dt + \sqrt{c X_{1}(t)} \, dW_{1}(t) \\ dX_{2}(t) = -b \left( X_{2}(t) - \beta_{2} \right) \, dt + \sqrt{c X_{2}(t)} \, dW_{2}(t), \end{array} \right.\] where \(\left( W_{1}(t) \right)_{t \geq 0}\) and \(\left( W_{2}(t) \right)_{t \geq 0}\) are independent standard Brownian motions. The parameters \(\beta_{1}\) and \(\beta_{2}\) are positive and not necessarily identical, while parameters \(b > 0\) and \(c > 0\), governing the speed of the mean reversion and volatility, are identical in both univariate CIRs. The motivation for defining the functional transformation of \(\left( X_{1}(t), X_{2}(t) \right)_{t \geq 0}\) comes from the relationship between two independent gamma distributions and the \(F\)-distribution. In particular, if \(X_{1} \sim \Gamma(\alpha \beta_{1}, \alpha)\) and \(X_{2} \sim \Gamma(\alpha \beta_{2}, \alpha)\) are independent, where \(\alpha = 2b/c\) and \(\beta_{2}, b, c > 0\), then the random variable \(Y = \beta_{2} X_{1}/\beta_{1} X_{2}\) has an \(F\) distribution with shape parameters \(2\alpha\beta_{1}\) and \(2\alpha\beta_{2}\), denoted by \(Y \sim F(2\alpha\beta_{1}, 2\alpha\beta_{2})\). The procedure for construction of time-changed \(F\)-diffusion is given in the following proposition.

Proposition. Let \(\left( X_{1}(t), X_{2}(t) \right)_{t \geq 0}\) be a two-dimensional CIR process with independent components given by SDE \((1)\), satisfying the condition \(\beta_{2} > (c/b) \max\{1, c/2\}\), and let \(\left( Y_{1}(t), Y_{2}(t) \right)_{t \geq 0}\) be its functional, given as follows: \[Y_{1}(t) = \displaystyle\frac{X_{1}(t)}{X_{2}(t)}, \quad Y_{2}(t) = X_{2}(t).\] If the time-change process \(\left( T(t) \right)_{t \geq 0}\) is defined as \[T(t) = \displaystyle\int_{0}^{t} \frac{1}{Y_{2}(s)} \, ds,\] then it has continuous inverse \((\tau_{t})_{t \geq 0}\) given by \[\tau_{t}(\omega) = \tau(t, \omega) = \inf{\{s \geq 0: T_{t} > t\}},\] and \(\left( Y_{1}(\tau_{t}) \right)_{t \geq 0}\) is \(F\)-diffusion driven by the Brownian motion \[\widetilde{W}^{*}_{1}(t) = \displaystyle\int_{0}^{\tau_{t}} \sqrt{\frac{1}{Y_{2}(s)}} \, d\widetilde{W}_{1}(s), \\ \widetilde{W}_{1}(t) = \sqrt{\frac{4 Y_{2}(t)}{Y_{1}(t) \left( Y_{1}(t) + 4 \right)}} \left(\sqrt{\frac{Y_{1}(t)}{Y_{2}(t)}} W_{1}(t) - \sqrt{\frac{Y_{1}^{2}(t)}{4 Y_{2}(t)}} W_{2}(t) \right),\] where \(\left( W_{i}(t) \right)_{t \geq 0}\), \(i = 1, 2\), are independent standard Brownian motions driving process \((1)\).

Bibliography

\([1]\) Avram, F., Leonenko, N., Šuvak, N. "On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions." Markov Process. Relat. Fields, vol. 19, no. 2, 2013, pp. 249-298.

\([2]\) Barndorff-Nielsen, O., Shiryaev, A. (2010). Change of Time and Change of Measure. World Scientific.

\([3]\) Gouriéroux, C., Jasiak, J. "Multivariate Jacobi process with application to smooth transitions." J. Econom., vol. 131, no. 1-2, 2006, pp 475-505.

\([4]\) Øksendal, B. "When is a stochastic integral a time change of a diffusion?" J. Theor. Probab. vol. 3, 1990, pp 207-226.

\([5]\) Šuvak N. "Time Scale Transformation in Bivariate Pearson Diffusions: A Shift from Light to Heavy Tails." Axioms., vol. 13, no. 11, 2024, pp. 765.

P8: Asymmetric branching trees and their genealogy

Frederik Mølkjær Andersen (University of Copenhagen)

stand: S8

Branching processes track population sizes over time and can be constructed by counting branches in subsets of an underlying family tree. Two tree structures have received particular attention: splitting trees, where reproduction occurs throughout an individual’s life, and branching trees, where reproduction happens only at death. Both satisfy the branching property, but in branching trees, it takes a particularly simple form that enables distributional characterizations via renewal-like integral equations.

We introduce a new class of family trees, called asymmetric branching trees, in which individuals may reproduce during life, but at each reproduction event, the mother is killed and then immediately resurrected as an additional offspring, inheriting key characteristics such as her age. In this way, offspring are still attached only at the ends of branches, preserving the structural simplicity of branching trees while capturing a broad range of biologically relevant models, including inhomogeneous age-dependent birth and death processes typically modeled with splitting trees.

Building on this foundation, we analyze the genealogical structure of asymmetric branching trees. The genealogy of an asymmetric branching tree, defined as the ancestral tree of extant individuals at a fixed observation time, admits a natural recursive description. With respect to an enlarged filtration, this genealogy satisfies a conditional branching property of its own, and by analyzing the reduced branching process, which counts only lineages with surviving descendants, we obtain explicit distributional results that fully characterize the genealogical structure.

P9: Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax.

Timofei Shashkov (University of Lausanne)

stand: S9

Abstract: Consider a bidimensional Brownian motion with independent components \[\bigl(B_{1}(t),B_{2}(t)\bigr), t\ge 0.\] For constants \(c_{1},c_{2}\in\mathbb{R}\) and reflection (tax) parameters \(\gamma_{1},\gamma_{2}\in[0,1]\), define the \(\boldsymbol\gamma\)-reflected process \[\bigl(X_{1}(t),X_{2}(t)\bigr) =\begin{pmatrix} B_{1}(t)-c_{1}t-\gamma_{1}\displaystyle\inf_{0\le s\le t}\!\bigl(B_{1}(s)-c_{1}s\bigr)\\[6pt] B_{2}(t)-c_{2}t-\gamma_{2}\displaystyle\inf_{0\le s\le t}\!\bigl(B_{2}(s)-c_{2}s\bigr) \end{pmatrix}, \qquad t\ge 0.\] In actuarial terms, \(c_{i}\) and \(\gamma_i\) correspond to the premium and tax rates of a line \(i\) respectively.

For a fixed time horizon \(T>0\) and a level ratio \(a \in \mathbb{R}\) we investigate the finite–time simultaneous ruin probability \[\mathbb{P}\left\{\exists\,t\in[0,T]: X_{1}(t)>u,\;X_{2}(t)>a\,u\right\}, \qquad u\to\infty,\] and derive its exact asymptotics as \(u\to\infty\).

P10: Asymptotic Behaviour of Vertex-Shift Dynamics on Unimodular Networks

Bharath Roy Choudhury (Dioscuri Centre for Random Walks in Geometry and Topology, Jagiellonian University)

stand: S10

Vertex-shifts provide a framework for navigating on graphs by specifying, for each vertex, a unique “next” vertex. Such a shift induces dynamics on the space \(\mathcal{G}_*\) of rooted networks: starting from a probability measure \(\mathcal{P}\) on \(\mathcal{G}_*\), one obtains the set \(\mathcal{A}_{\mathcal{P}}\) of push-forward measures under the successive iterations of the vertex-shift. The accumulation points of \(\mathcal{A}_{\mathcal{P}}\), termed as \(f\)-probabilities, describe the long-term behaviour of these iterations. This paper establishes general sufficient conditions on \(f\)—namely periodicity and finiteness of its orbits—for the existence of \(f\)-probabilities. Furthermore, when \(f\)-probabilities exist, we give sufficient conditions under which they are either absolutely continuous or singular with respect to the initial measure \(\mathcal{P}\). In the special case where \(f\) is \(1\)-periodic and \(\mathcal{P}\) is unimodular, we derive the Radon-Nikodym derivative of the \(f\)-probability with respect to \(\mathcal{P}\).

Applications include the Parent vertex-shift on certain classes of unimodular Family Trees—where each vertex is mapped to its parent—yielding an explicit, unique \(f\)-probability. A second application considers the record vertex-shift on skip-free-to-the-left (also known as left-continuous) random walks with non-negative mean increments. In this setting, we derive the record probability, namely the \(f\)-probability associated to the record vertex-shift that maps each point on the trajectory of the random walk to its next record. The record probability describes the trajectory “seen from” the maximum value of the walk. These results unify dynamical and measure-theoretic perspectives on network navigation, extending tools from stochastic geometry and unimodular random graphs.

Some of the main results are as follows.

Theorem. Let \([\mathbf{G},\mathbf{o}]\) be a random network, \(\mathcal{P}\) be its distribution and \(f\) be an a.s. \(1\)-periodic vertex-shift on \([\mathbf{G},\mathbf{o}]\). Then, the \(f\)-probability \(\mathcal{P}^f\) of \(\mathcal{P}\) exists.

Moreover, if \([\mathbf{G},\mathbf{o}]\) is unimodular, then \(\mathcal{P}^f\) is absolutely continuous with respect to \(\mathcal{P}\). The limit function \(d^f_{\infty}(o)= \lim_{n \rightarrow \infty}d^f_n(o)\) exists and is almost surely finite under \(\mathcal{P}^f\). The Radon-Nikodym derivative of \(\mathcal{P}^f\) with respect to \(\mathcal{P}^f\) is given by \[\frac{d \mathcal{P}^f}{d \mathcal{P}}[G,o] = d_{\infty}(o) \mathbf{1}\{\text{o is a trap}\}.\]

Theorem. Let \([\mathbb{Z},0,X]\) be the marked rooted network whose vertices are \(\mathbb{Z}\), root is \(0\), and marks \(X=(X_n)_{n \in \mathbb{Z}}\) on edges \((n,n+1)_{n \in \mathbb{Z}}\) represent the increments of a skip-free-to-the-left random walk with \(\mathbb{E}[X_0] \geq 0\). Define the record vertex-shift \(R\) by \(R(i) = \inf\{j>i: \sum_{k=j}^{i-1}X_k \geq 0\}\) for all \(i \in \mathbb{Z}\). Then, the record probability \(\lim_{n \to \infty} [\mathbb{Z},R^n(0),X]\) exists. Moreover, this limit measure admits an explicit description via an encoding of a random Family Tree.

Bibliography

\([1]\) Bharath Roy Choudhury. Records of Stationary Processes and Unimodular Graphs. ArXiv, doi:10.48550/arXiv.2312.08121.

\([2]\) François Baccelli and Bharath Roy Choudhury. Genealogies of Records of Stochastic Processes with Stationary Increments as Unimodular Trees. ArXiv, doi:10.48550/arXiv.2403.05657.

\([3]\) François Baccelli, Mir-Omid Haji-Mirsadeghi, and Ali Khezeli. Eternal Family Trees and Dynamics on Unimodular Random Graphs. In Contemporary Mathematics, American Mathematical Society, Vol. 719, 2018, pp 85-127. doi:10.1090/conm/719/14471.

\([4]\) François Baccelli and Mir-Omid Haji-Mirsadeghi. Point-Map-Probabilities of a Point Process and Mecke’s Invariant Measure Equation. The Annals of Probability, Vol. 45, no. 3, 2017, pp 1723-1751.

P11: Estimates of kernels and ground states for Schrödinger semigroups

Miłosz Baraniewicz (Wroclaw University of Science and Technology)

stand: S11

We consider the Schrödinger operator of the form \(H=-\Delta+V\) acting in \(L^2(\mathbb{R}^d,dx)\), \(d \geq 1\), where the potential \(V:\mathbb{R}^d \to [0,\infty)\) is a locally bounded function. The corresponding Schrödinger semigroup \(\big\{e^{-tH}: t \geq 0\big\}\) consists of integral operators, i.e. \[e^{-tH} f(x) = \int_{\mathbb{R}^d} u_t(x,y) f(y) dy, \quad f \in L^2(\mathbb{R}^d,dx), \ t>0.\]

In first part of the talk i will present new estimates for heat kernel of \(u_t(x,y)\). Our results show the contribution of the potential is described separately for each spatial variable, and the interplay between the spatial variables is seen only through the Gaussian kernel.

This estimates will be presented on two common classes of potentials. For confining potentials we get two sided estimates and for decaying potentials we get new upper estimate.

Methods we used to estimated kernel of semigroup allow to easily obtain sharp estimates of ground state for slowly varying potentials.

Bibliography

\([1]\) M. Baraniewicz, K. Kaleta. "Integral kernels of Schrödinger semigroups with nonnegative locally bounded potentials." Studia Mathematica, 275, 2024, 147-173.

\([2]\) M. Baraniewicz. "Estmates of ground state for classical Schrödinger operator." Probability and Mathematical Statistics, 44, 2025, 267-277.

P12: Intrinsic ultracontractivity of Feynman-Kac semigroups for cylindrical stable processes

Kinga Sztonyk (Wrocław University of Science and Technology)

stand: S12

The following Schrödinger operator \[K = K_0 + V,\] where \[K_0 = \sqrt{-\frac{\partial^2}{\partial x_1^2}} + \sqrt{-\frac{\partial^2}{\partial x_2^2}}\] is an example of a nonlocal, anisotropic, singular Lévy operator. We consider potentials \(V : \mathbb{R}^2 \to \mathbb{R}\) such that \(V(x)\) goes to infinity as \(|x| \to \infty\). The operator \(-K_0\) is a generator of a process \(X_t = (X_t^{(1)}, X_t^{(2)})\), sometimes called cylindrical, such that \(X_1^{(1)}\), \(X_2^{(2)}\) are independent symmetric Cauchy processes in \(\mathbb{R}\).

We define the Feynman-Kac semigroup \[T_t f(x) = E^x \left( \exp \left( -\int_0^t V(X_s) \, ds \right) f(X_t) \right).\] Operators \(T_t\) are compact for every \(t > 0\). There exists an orthonormal basis \(\{ \phi_n \}_{n=1}^{\infty}\) in \(L^2 (\mathbb{R}^2)\) and a corresponding sequence of eigenvalues \(\{\lambda_n \}_{n=1}^{\infty}\), \(0<\lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \dots\), \(\lim_{n \to \infty} \lambda_n = \infty\) such that \(T_t \phi_n = e^{-\lambda_n t} \phi_n\). We can assume that \(\phi_1\) is positive and continuous on \(\mathbb{R}^2\). The main result I would like to present concerns estimates for \(\phi_1\) and intrinsic ultracontractivity of the semigroup \(T_t\) under certain conditions on the potential \(V\).

Bibliography

\([1]\) Tadeusz Kulczycki, Kinga Sztonyk. "Intrinsic ultracontractivity for Schrödinger semigroups based on cylindrical fractional Laplacian on the plane." arXiv:2407.14325.

P13: Path-dependent option pricing with two-dimensional PDE using MPDATA

Paweł Magnuszewski (AGH University of Krakow)

stand: S13

In this poster, we discuss a simple yet robust PDE method for evaluating path-dependent Asian-style options using the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result, all terms of the Black-Merton-Sholes equation are consistently represented using a single high-order numerical scheme for the advection operator. We detail the additional steps required to solve the two-dimensional valuation problem compared to MPDATA valuations of vanilla instruments documented in a prior study. Using test cases employing fixed-strike instruments, we validate the solutions against Monte Carlo valuations, as well as against an approximate analytical solution in which geometric instead of arithmetic averaging is used. The analysis highlights the critical importance of the MPDATA corrective steps that improve the solution over the underlying first-order "upwind" step. The introduced valuation scheme is robust: conservative, non-oscillatory, and positive-definite; yet lucid: explicit in time, engendering intuitive stability-condition interpretation and inflow/outflow boundary-condition heuristics. MPDATA is particularly well suited for two-dimensional problems as it is not a dimensionally split scheme. The documented valuation workflow also constitutes a useful two-dimensional case for testing advection schemes featuring both Monte Carlo solutions and analytic bounds. An implementation of the introduced valuation workflow, based on the PyMPDATA package and the Numba Just-In-Time compiler for Python, is provided as free and open source software.

P14: The distributions of the mean of random vectors with fixed marginal distribution

Jacques Labuschagne (UNIWERSYTET ŁÓDZKI)

stand: S14

This poster will be based upon our publication, in which we used the recent results concerning the non-uniqueness of the center of the mix for completely mixable probability distributions we obtain the following result: For each \(d\in\mathbb N\) and each non-empty bounded Borel set \(B\subset\mathbb R^d\) there exists a \(d\)-dimensional probability distribution \(\boldsymbol\mu\) satisfying what follows: for each \(n\geq3\) and each probability distribution \(\boldsymbol\nu\) on \(B\) there exist \(d\)-dimensional random vectors \(\mathbf X_{\boldsymbol\nu,1},\mathbf X_{\boldsymbol\nu,2},\dots,\mathbf X_{\boldsymbol\nu,n}\) such that \(\frac1n(\mathbf X_{\boldsymbol\nu,1}+\mathbf X_{\boldsymbol\nu,2}+\dots+\mathbf X_{\boldsymbol\nu,n})\sim\boldsymbol\nu\) and \(\mathbf X_{\boldsymbol\nu,i}\sim\boldsymbol\mu\) for \(i=1,2,\dots,n\). We also show that the assumption regarding the boundedness of set \(B\) cannot be completely omitted, but it can be substantially weakened.

Bibliography

\([1]\) A. Komisarski, J. Labuschagne, The Distributions of the Mean of Random Vectors with Fixed Marginal Distribution, J. Theor. Probab. (2023), https://doi.org/10.1007/s10959-023-01277-2