We indicate a gap in the construction of the generalized Skorokhod representation given in Blackwell and Dubins ( PAMS , 89 (1983) , 691–692). It is shown that the construction, taken without correction, leads to false statements.

Let ( X , F , μ ) (\mathcal {X},\mathcal {F},\mu ) and ( Y , G , ν ) (\mathcal {Y},\mathcal {G},\nu ) be probability spaces and ( Z n ) (Z_n) be a sequence of random variables with values in ( X × Y , F ⊗ G ) (\mathcal {X}\times \mathcal {Y},\,\mathcal {F}\otimes \mathcal {G}) . Let Γ ( μ , ν ) \Gamma (\mu ,\nu ) be the collection of all probability measures p p on F ⊗ G \mathcal {F}\otimes \mathcal {G} such that p ( A × Y ) = μ ( A ) and p ( X × B ) = ν ( B ) for all A ∈ F and B ∈ G . \begin{equation*} p\bigl (A\times \mathcal {Y}\bigr )=\mu (A)\quad \text {and}\quad p\bigl (\mathcal {X}\times B\bigr )=\nu (B)\quad \text {for all }A\in \mathcal {F}\text { and }B\in \mathcal {G}. \end{equation*} In this paper, we build some probability measures Π \Pi on Γ ( μ , ν ) \Gamma (\mu ,\nu ) . In addition, for each such Π \Pi , we assume that ( Z n ) (Z_n) is exchangeable with de Finetti’s measure Π \Pi and we evaluate the conditional distribution Π ( ⋅ ∣ Z 1 , … , Z n ) \Pi (\,\cdot \mid Z_1,\ldots ,Z_n) . In Bayesian nonparametrics, if ( Z 1 , … , Z n ) (Z_1,\ldots ,Z_n) are the available data, Π \Pi and Π ( ⋅ ∣ Z 1 , … , Z n ) \Pi (\,\cdot \mid Z_1,\ldots ,Z_n) can be regarded as the prior and the posterior, respectively. To support this interpretation, it suffices to think of a problem where the unknown probability distribution of some bivariate phenomenon is constrained to have marginals μ \mu and ν \nu . Finally, analogous results are obtained for the set Γ ( μ ) \Gamma (\mu ) of those probability measures on F ⊗ G \mathcal {F}\otimes \mathcal {G} with marginal μ \mu on F \mathcal {F} (but arbitrary marginal on G \mathcal {G} ). That is, we introduce some priors on Γ ( μ ) \Gamma (\mu ) and we evaluate the corresponding posteriors.

This paper introduces a new type of nonlinear shrinkage estimators for the precision matrix in high-dimensional settings, where the dimension of the data-generating process exceeds the sample size. The proposed estimators incorporate the Moore–Penrose inverse and the ridge-type inverse of the sample covariance matrix, and they include linear shrinkage estimators as special cases. Recursive formulae of these higher-order nonlinear shrinkage estimators are derived using partial exponential Bell polynomials. Through simulation studies, the new methods are compared with the oracle nonlinear shrinkage estimator of the precision matrix for which no analytical expression is available.

A long time ago, it was shown that random series representations of fractional Brownian motion (fBm) generated by smooth orthonormal wavelet bases of L 2 ( R ) L^2(\mathbb {R}) are optimal, in the sense that they converge at the best possible rate provided by the sequence of the ℓ \ell -approximation numbers of this process. The goal of our article is to extend this optimality result to the random series representation of fBm generated by the Haar basis of L 2 ( R ) L^2(\mathbb {R}) , which offers the advantage to provide explicit results and simple computations. One of the main challenges in this context is that, in contrast to smooth wavelet functions, fractional primitives of the discontinuous Haar functions are badly localized. In order to overcome it, we will use a new strategy relying on Abel transform.

The simplest multivariate trigonometric model of symmetric textured surface is considered, that is observed on the background of homogeneous and isotropic Gaussian, in particular, strongly dependent random field on R M \mathbb {R}^M , M ≥ 3 M\geq 3 . In the specified regression model strong consistency and asymptotic normality of periodogram estimates of amplitude and angular frequencies are proved.

Consider two balls with radius r ∈ ( 0 , 1 ) r\in (0,1) whose centers are at a distance 2, positioned symmetrically with respect to the origin in R d {\mathbb R}^d . Suppose that the initial velocities are independent standard normal vectors. We prove that when r → 0 r\to 0 , the collision probability goes to 0 as r d − 1 r^{d-1} , and the asymptotic collision location distribution is a (defective) t t -distribution. This distribution is rotationally symmetric about the origin for no apparent reason.

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.

The paper considers equations driven by processes with zero cubic variation in probability. The integral with respect to such processes is a generalization of the symmetric integral with respect to stochastic measures. The existence and uniqueness of the solutions are obtained. It is proved that the convergence of the functions in these equations implies the convergence of the solutions.

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if f f is a Besicovitch almost periodic function and V V is a random variable uniformly distributed on [ − 1 , 1 ] [-1,1] , then the random variables f ( L ⋅ V ) f(L\cdot V) converge in distribution, as L → ∞ L\to \infty , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes ( f ( L ⋅ V + t ) ) t ∈ R (f(L\cdot V+t))_{t\in \mathbb {R}} in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. Further we investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of Akbary, Ng and Shahabi (2014) and earlier works.

We investigate the Gatheral model of double mean-reverting stochastic volatility, in which the drift term itself follows a mean-reverting process, and the overall model exhibits mean-reverting behavior. We demonstrate that such processes can attain values arbitrarily close to zero and remain near zero for extended periods, making them practically and statistically indistinguishable from zero. To address this issue, we propose a modified model incorporating Skorokhod reflection, which preserves the model’s flexibility while preventing volatility from approaching zero.