Laplace’s method is used to approximate intractable integrals in a statistical problems. The relative error rate of the approximation not worse than Op(n−1). We provide first lower bounds showing that n−1 tight.

A = (ank) be a regular summability matrix. In the present paper we deal with subspaces of space A?statistically convergent sequences obtained by rate at which A?statistical limit tends to zero. We prove that sequence is A?strongly if and only it A?uniformly integrable o (an) where positive nonincreasing sequence. also make link between A?strong convergence A?distributional (an). Finally, as an ...

Given a set P of n points in R d , a fundamental problem in computational geometry is concerned with nding the smallest enclosing \range" of P. Well known instances of this problem include nding, e.g., the smallest enclosing box 10], smallest enclosing simplex 11], minimum volume ball 17, 20], (2-D) smallest enclosing ellipsoid 16, 20, 2, 14], and minimum volume annulus 1, 8, 5]. In this paper,...

We present a simple method to deal with caustics in the semiclassical approximation to the partition function of a one-dimensional quantum system. The procedure, which makes use of complex trajectories, is applied to the quartic double-well potential.