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We write this note to correct [1, (6.9), (6.13), (7.1), (7.2)] because there was one term missed in [1, (6.9)].

Stein’s method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut’s formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein’s exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance. We apply the theorem to the unadjusted Langevin algorithm.

Let $\xi=(\xi_i, 1 \leq i \leq n)$ and $\eta= (\eta_i, 1 \leq i \leq n)$ be standard normal random variables with covariance matrices $R^1=(r_{ij}^1)$ and $R^0=(r_{ij}^0)$, respectively. Slepian's lemma says that if $r_{ij}^1 \geq r_{ij}^0$ for $1 \leq i, j \leq n$, the lower bound $\P(\xi_i \leq u \mb{for} 1 \leq i \leq n ) /\P(\eta_i \leq u \mb{for} 1 \leq i \leq n ) $ is at least $1$. In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Revesz, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues.

Let B 0,B 1, ⋯ ,B n be independent standard Brownian motions, starting at 0. We investigate the tail of the capture time where 0

This paper is part of our efforts to develop Stein's method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Stein's method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments.

Lower bounds on the small ball probability are given for Brownian sheet type Gaussian fields as well as for general Gaussian fields with stationary increments in ℝ d . In particular, a sharp bound is found for the fractional Levy Brownian fields.

A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.

Let {X(t),t ≧ 0} be a stationary Gaussian process withEX(t)=0,EX 2(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t |α + o (|t|α)ast→0 for someC>0, 00 and (iii) supt≥s|r(t)| 0. Put ξ(t)= sup {s:0 ≦s ≦t,X(s)≧ (2logs)1/2}. The law of the iterated logarithm implies $$lim sup_{t \to \infty } (\xi (t) - t) = 0$$ a.s. This paper gives the lower bound of ζ(t) and obtains an Erdős-Revesz type LIL, i.e., $$\lim \inf _{t \to \infty } (\xi (t) - t)/(t(\log t)^{{{(\alpha - 2)} \mathord{\left/ {\vphantom {{(\alpha - 2)} {(2}}} \right. \kern-\nulldelimiterspace} {(2}}\alpha )} \log \log t) = - (2 + \alpha )\sqrt \pi /(\alpha {\rm H}_\alpha (2C)^{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} )$$ a.s. if 0