and its convergence to 1/b. We give two proofs of explicit error bounds for this convergence, one using Stein's method for translated Poisson approximation and one using Fourier analysis. The error bound using Fourier analysis yields exponentially decaying error bounds for fixed b, which generalizes the already known case b = 2; however, it makes use of a special representation due to Tanny (1973). In contrast, Stein's method only yields polynomially decaying error bounds, but we hope it has potential for generalization beyond the present setting.
We present an elementary proof that the asymptotic entropy of a random walk on a countable abelian group is zero when the entropy of the first step of the random walk is finite. Unlike the traditional proof, our approach does not rely on the boundary theory of random walks. To our best knowledge, our direct proof is new even for the group of integers.
We consider the problem of constructing sharp upper and lower bounds for the second moment of sums of arbitrary, square-integrable, dependent random variables, ∑i=1ndi, using recent results in the theory of decoupling inequalities. Specifically, we give a new proof for the complete decoupling inequality, which provides a lower bound for the sum of dependent square-integrable nonnegative random variables 1 2E∑i=1nz i2≤E∑i=1nd i2, where zi= Ldi for all i≤n and zi’s are mutually independent. We further develop sharp tangent decoupling inequalities for the variance and second moment of arbitrary square-integrable sequences {di}1≤i≤n, without requiring nonnegativity: Var∑i=1nd i≤2Var∑i=1ne i, and E∑i=1nd i2≤2E∑i=1ne i2−E∑i=1ne i2, where {ei} is the decoupled sequences of {di}. These results sharpen and generalize previous bounds for randomly stopped sums and martingale-type sequences. Applications include refined Chebyshev and Paley-Zygmund-type inequalities under minimal moment assumptions. The inequalities presented here offer new tools for probabilistic analysis in both classical and modern settings involving dependent structures.
The general isomorphism theorem for transient, non necessarily symmetric, Markov processes conditioned to start and to die at the same point, established by Eisenbaum and Kaspi (2009) has been derived by Fitzsimmons and Rosen (2014) from Palm's disintegration formula for loop soups. By allowing Markov processes to start and to end at different points, the isomorphism theorem of Dynkin (1983) should not a priori be obviously connected to loop measures. Nevertheless we show here that Dynkin's isomorphism theorem can be seen as a consequence of the disintegration formula for loop soups. Finally we show that the used arguments are actually available in the large framework of positive infinitely divisible processes and lead similarly to isomorphism theorems.
This paper is the last in a series devoted to constructing stochastic motions representing the two-dimensional N-body delta-Bose gas for all integers N >= 3 via Feynman-Kac-type formulas; see [2, 3, 4] for the earlier parts of this series. The main result here supplements [2, 3] by establishing an explicitly defined bijective transformation between two classes of Langevin-type SDEs with general coefficients of a particular form. In essence, this transformation provides an SDE-based realization of the principle that the dynamics of the relative motions among N particles and of the center of mass uniquely determine the overall dynamics of the particles.
This work investigates continuous dependence on initial values for 1-dimensional (distribution-dependent) SDEs with H & ouml;lder continuous coefficients. The regularity obtained in this work coincides with the regularity associated with SDEs with Lipschitz continuous coefficients, and hence is sharpest. Our approach depends on an application of Meyer-Tanaka's formula and a subtle estimation of local times for continuous semimartingales.
Let (Xn) be a sequence of random variables with values in a standard Borel space S. We investigate the condition E{f(Xn+1) X1,. . . ,Xn} converges in probability, (*) as n H co, for each bounded Borel function f : S H R. Some consequences of (*) are highlighted and various sufficient conditions for it are obtained. In particular, (*) is characterized in terms of stable convergence. It is also shown that, under (*), there is a random probability measure alpha on S such that E{f(Xn+1) X1,. . . , Xn}-H P f f d alpha for each bounded Borel f. Moreover, since (*) holds whenever (Xn) is conditionally identically distributed, three weak versions of the latter condition are investigated. For each version, our main goal is proving (or disproving) that (*) holds. Several counterexamples are given as well.
The critical beta-splitting tree is a random rooted tree, in which a set of m >= 2 leaves is recursively split into two subsets containing i and m i leaves with probabilities proportional to i-1(m i)-1. We further explore a connection initially unveiled in Iksanov (2025) between the critical beta-splitting tree and an infinite balls-in-boxes scheme. Using this connection, we derive a new joint central limit theorem for components of the height of a leaf chosen uniformly at random in the discrete version of the critical beta-splitting tree. Also, we obtain a joint central limit theorem for the heights in the discrete and continuous versions of the critical beta-splitting tree.
We show that a site percolation is a stronger model than a bond percolation. We use the van den Berg-Kesten (vdBK) inequality to prove that site percolation on a neighborhood of a vertex of degree 4 cannot be simulated even approximately by bond percolation, and develop a decision tree technique to prove the same for a neighborhood of a vertex of degree 3. This technique can be used to obtain inequalities for connectedness probabilities, including a conjectured inequality of Erik Aas.
We study the effect of local perturbations on the recurrence of random walks with long jumps. Such walks serve as discrete models for infinite-horizon Lorentz processes, in which a particle can take arbitrarily long steps in specific directions. Motivated by a question of Sinai in the finite-horizon case and its extension by Sz & aacute;sz to the infinite-horizon setting, we give recurrence and transience criteria for long-jump walks on Z2 and certain classes of graphs, and we prove that local perturbations in a bounded region do not change the recurrence property. Our proofs combine the Markov chain approach with the electrical network method, making the arguments transparent to a broad audience in probability.
We establish existence of probabilistically strong solutions and pathwise uniqueness for a class of quasilinear stochastic evolution equations on bounded domains. Our results combine recent weak existence results for quasilinear stochastic evolution equations in an L^p-setting (with p > 2) with Yamada–Watanabe theory. To establish pathwise uniqueness, we rely on an L^1-contraction argument.
We introduce a novel notion of divergence between continuous martingales; the reciprocal specific relative entropy. First, we motivate this definition from multiple perspectives. Thereafter, we solve the reciprocal specific relative entropy minimization problem over the set of win-martingales (used as models for prediction markets [2]). Surprisingly, we show that the optimizer is the renowned neutral Wright-Fisher diffusion. We also justify that this diffusion is in a sense the most salient win-martingale, since it is uniquely selected when we suitably perturb the degenerate martingale optimal transport problem of variance minimization.
In this short note, we provide an explicit sufficient condition for non-explosion of Crump-Mode-Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is - at least for rate sequences without excessive oscillations - remarkably close to being necessary. At the same time, it is not necessary in full generality: we construct a counterexample which also yields a general preferential attachment tree without fitness with an infinite path and no vertices of infinite degree, thereby answering an open question previously raised in the literature.
We consider a reflected process in the positive orthant driven by an exogenous c & agrave;dl & agrave;g process. For a given input process, we show that solutions to the reflection problem can be continued at a jump time if and only if the proposed jump of the unregulated process is within the dual cone of a linear programming problem associated with the state of the system. As a consequence, for piecewise non-decreasing driving processes there exists a unique minimal strong solution to the given particle system up until the stopping time at which such a non-allowed jump first occurs. We apply this model to study the ruin of interconnected insurance firms, where the stopping time can be interpreted as the failure of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to a class of L & eacute;vy processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.
The Fr & eacute;chet-Shohat Theorem (FST) (1931) is a well-established result concerning determinate moment problems. We endeavor to address an analogous problem within the realm of indeterminate Hamburger and Stieltjes moment problems, focusing exclusively on absolutely continuous random variables. We demonstrate that, under an additional condition stipulating the convergence of the entropy of a sequence of distribution functions to the entropy of the unique maximum entropy distribution, a stronger mode of convergence is achieved, which subsequently implies convergence in distribution. This result is attainable due to the inherent property of indeterminate moment problems possessing a unique density ghmax, distinguishable from other solutions by its maximal entropy. In conclusion, the moment convergence implies weak convergence under determinacy and under indeterminacy, if there is entropy convergence to the ghmax then there is also weak convergence.
The mixed fractional Brownian motion-the sum of independent fractional and standard Brownian motions-is known to be a semimartingale if the Hurst exponent H of its fractional component satisfies H > 3/4. The question posed in the title is motivated by recent findings in quantitative finance. In this note, we show that the drift in its Doob-Meyer decomposition has a derivative that is gamma-H & ouml;lder continuous for any gamma < 2H-3/2.
We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglb & ouml;ck, Huesmann and K & auml;llblad.
We consider a stationary Poisson process of k-planes in the d-dimensional hyperbolic space H-d of constant curvature -1, with d >= 4 and 1 <= k <= d-1. It is known that, after centring and normalization, the total k-volume of all intersections of k-planes with a geodesic ball of radius R converges in distribution, as R -> infinity, to a non-Gaussian infinitely divisible random variable Z(d,k) whenever 2k > d + 1. We investigate the distributional behaviour of Z(d,k) in the high-dimensional regime d -> infinity and depending on how fast k grows in relation to d. We derive precise conditions for the variance normalized sequence to converge in law to a standard Gaussian random variable or to a degenerate law, respectively, and show that an alternative rescaling of the Levy measures yields an explicit non-Gaussian infinitely divisible limit for fixed codimension d - k and a standard Gaussian limit for d - k -> infinity.
For an indeterminate Hamburger moment problem we consider an infinite family of analytic densities solving the moment problem and we prove that they all have finite (Shannon) entropy. These densities are either all bounded or all unbounded. The result is illustrated by the Al-Salam-Carlitz moment problem, where all the densities in the family are bounded.
In this short note, we characterize stability of the Kim–Milman flow map – also known as the probability flow ODE – with respect to variations in the target measure. Rather than the Wasserstein distance, we show that stability holds with respect to the relative Fisher information
