Explore the vast range of titles available.

Practical Bayesian statistics with realistic models usually gives posterior distributions that are analytically intractable, and inferences must be made via numerical integration. In many cases, the integrands can be transformed into periodic functions on the unit d-dimensional cube, for which quasirandom sequences are known to give efficient numerical integration rules. This paper reviews some relevant theory, defines new criteria for identifying suitable quasirandom sequences and suggests some extensions to the basic integration rules. Various quasirandom methods are then compared on the sort of integrals that arise in Bayesian inference and are shown to be much more efficient than Monte Carlo methods.

For a class of expansive transformations of the unit interval, we show a local limit theorem for the process (f ⚬ Tn)n ∈ N, where f is a real bounded variation function. We show also, that the speed in the central limit theorem is$1/\\sqrt n$.

The lead digit behavior of a large class of arithmetic sequences is determined by using results from the theory of uniform distribution $\\operatorname{mod} 1$. Theory for triangular arrays is developed and applied to binomial coefficients. A conjecture of Benford's that the distribution of digits in all places tends to be nearly uniform is verified.

An explicit construction of invariant measures for a certain class of continuous state Markov processes is presented. A special version of these processes is of interest in the theory of representation of real numbers ($\\beta$-expansions). Previous results of Renyi and Parry are generalized, and an open problem of Parry is resolved.

Let $\\{n_k, k \\geqq 1\\}$ be a sequence of random variables uniformly distributed over $\\{0, 1\\}$ and let $F_N(t)$ be the empirical distribution function at stage $N$. Put $f_n(t) = N(F_N(t) - t)(N\\log\\log N)^{-\\frac{1}{2}}, 0 \\leqq t \\leqq 1, N \\geqq 3$. For strictly stationary sequences $\\{n_k\\}$ where $n_k$ is a function of random variables satisfying a strong mixing condition or where $n_k = n_k x \\mod 1$ with $\\{n_k, k \\geqq 1\\}$ a lacunary sequence of real numbers a functional law of the iterated longarithm is proven: The sequence $\\{f_N(t), N \\geqq 3\\}$ is with probability 1 relatively compact in $D\\lbrack 0, 1\\rbrack$ and the set of its limits is the unit ball in the reproducing kernel Hilbert space associated with the covariance function of the appropriate Gaussian process.