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In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $$\alpha ,\beta \rightarrow \infty $$ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces $${H}_d(\mathbb F )$$ with growing dimensions $$d$$ over the fields $$\mathbb F =\mathbb R ,\ \mathbb C $$ or the quaternions $$\mathbb H $$ .

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries.

This paper deals with moduli of continuity for paths of random processes indexed by a general metric space $$\Theta $$ with values in a general metric space $${{\mathcal {X}}}$$ . Adapting the moment condition on the increments from the classical Kolmogorov–Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric space $${{\mathcal {X}}}$$ is complete. This result is universal in the sense that its applicability depends only on the geometry of the space $$\Theta $$ . In particular, it is always applicable if $$\Theta $$ is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result, a criterion is presented to guarantee uniform tightness of random processes with continuous paths. This is applied to find central limit theorems for Banach-valued random processes.

We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t ‖X(s)‖ of an ℝ d -valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.

We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, we introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this we mean a sequence of controls for which the payoff functions approach the value function.

We show that the conditional central limit theorem can take place for a stationary process defined on a nonergodic dynamical system while this last does not satisfy the central limit theorem for any ergodic component. There exists an ergodic Markov chain such that the conditional central limit theorem is satisfied for an invariant measure but fails to hold for almost all starting points.

We provide exact asymptotics for the tail probabilities $${\mathbb {P}}\{ S_{n,r} > x \}$$ as $$x \rightarrow \infty $$ , for fixed n, where $$S_{n,r}$$ is the r-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the r-trimmed limits.

In this paper, we consider a large class of super-Brownian motions in $${\mathbb {R}}$$ with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval $$(-\delta t,\delta t)$$ for $$\delta >0$$ . The growth rate is given in terms of the principal eigenvalue $$\lambda _{1}$$ of the Schrödinger-type operator associated with the branching mechanism. From this result, we see the existence of phase transition for the growth order at $$\delta =\sqrt{\lambda _{1}/2}$$ . We further show that the super-Brownian motion shifted by $$\sqrt{\lambda _{1}/2}\,t$$ converges in distribution to a random measure with random density mixed by a martingale limit.

We consider perturbed empirical distribution functions $$\hat F_n (x) = 1/n\sum\nolimits_{i = 1}^n {G_n (x - X_i )} $$ , where {Ginn, n≥1} is a sequence of continuous distribution functions converging weakly to the distribution function of unit mass at 0, and {Xi, i≥1} is a non-stationary sequence of absolutely regular random variables. We derive the almost sure representation and the law of the iterated logarithm for the statistic $$\hat F_n (U_n )$$ whereUn is aU-statistic based onX1,...,Xn. The results obtained extend or generalize the results of Nadaraya,(7) Winter,(16) Puri and Ralescu,(9,10) Oodaira and Yoshihara,(8) and Yoshihara,(19) among others.