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We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $$H>\frac{1}{2}$$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $$L^1$$ -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $$n^{1-2H}$$ , which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

A random permutation ofN items generated by a sequence ofK random transpositions is considered. The method of strong uniform times is used to give an upper bound on the variation distance between the distributions of the random permutation generated and a uniformly distributed permutation. The strong uniform time is also used to find the asymptotic distribution of the number of fixed points of the generated permutation. This is used to give a lower bound on the same variation distance. Together these bounds give a striking demonstration of the threshold phenomenon in the convergence of rapidly mixing Markov chains to stationarity.

We prove singularity of some distributions of random continued fractions that correspond to iterated function systems with overlap and a parabolic point. These arose while studying the conductance of Galton-Watson trees.

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.

Let X be a symmetric Lévy process with $$Ee^{i\lambda X_t } = e^{ - t\psi (\lambda )}$$ Let $$\phi (x) = \frac{2}{\pi }\int_0^\infty {\frac{{1 - \cos \lambda x}}{{\psi (\lambda )}}} d\lambda $$ Assume that • ψ(λ) is regularly varying at zero with index 1<α≤2 and (1/ψ(λ)) I [λ≥1]∈L 1(R). • φ(x) is increasing on [0, ∞)Let L x t denote the local time of X at x up to time t. Following The most visited sites of symmetric stable processes, by Bass, Eisenbaum, and Shi, let V(t) be such that L V(t) t =sup x∈R L x t . We call V(t) the most visited site of X up to time t. We show that under the above conditions on X,V(t) is transient. In particular, for all γ>9 $$\mathop {\lim }\limits_{t \to \infty } \frac{{\left| {V(t)} \right|}}{{\phi ^{ - 1} \left( {\frac{{t\psi ^{ - 1} (1/t)}}{{(\log t)^\gamma }}} \right)}} = \infty {\text{ a}}{\text{.s}}{\text{.}}$$ This result is obtained for symmetric stable processes in the above reference. We use their approach and many of their methods.

A time-space harmonic polynomial for a stochastic process M=(M t) is a polynomial P in two variables such that P(t, M t) is a martingale. In this paper, we investigate conditions for the existence of such polynomials of each degree in the second, “space,” argument. We also describe various properties a sequence of time-space harmonic polynomials may possess and the interaction of these properties with distributional properties of the underlying process. Thus, continuous-time conterparts to the results of Goswami and Sengupta,(2) where the analoguous problem in discrete time was considered, are derived. A few additional properties are also considered. The resulting properties of the process include independent increments, stationary independent increments and semi-stability. Finally, a generalization to a “measure” proposed by Hochberg(3) on path space is obtained.