This short note investigates convergence of adaptive Markov chain Monte Carlo algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly. We focus on the containment condition introduced Roberts and Rosenthal (2007). We show that if the containment condition is not satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then any nonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail, and conclude that they should not be used.

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

Consider a sequence of Markov-dependent trials where each trial produces a letter of a finite alphabet. Given a collection of patterns, we look at this sequence until one of these patterns appears as a run. We show how the method of gambling teams can be employed to compute the probability that a given pattern is the first pattern to occur.

We look at joint regular variation properties of MA(∞) processes of the form X = (Xk, k ∈ Z), where Xk = ∑j=0∞ψjZk-j and the sequence of random variables (Zi, i ∈ Z) are independent and identically distributed with regularly varying tails. We use the setup of MO-convergence and obtain hidden regular variation properties for X under summability conditions on the constant coefficients (ψj: j ≥ 0). Our approach emphasizes continuity properties of mappings and produces regular variation in sequence space.

The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, π). To fix the scale, one side chosen arbitrarily is assigned unit length; let a and b denote the lengths of the other sides. We find the density functions of a / b, max{a, b}, min{a, b}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, ∞) and (½, ∞), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).

In this note we provide a simple alternative probabilistic derivation of an explicit formula of Kwan and Yang (2007) for the probability of ruin in a risk model with a certain dependence between general claim interoccurrence times and subsequent claim sizes of conditionally exponential type. The approach puts the type of formula in a general context, illustrating the potential for similar simple ruin probability expressions in more general risk models with dependence.

We obtain the long-time behaviour to the variance of the distribution process associated with the solution of the telegraph equation. To this end, we use a version of the Karamata-Feller Tauberian theorem.

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial/final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination/source. Suppose further that connections are established using ‘near geodesics’, constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In Kendall (2011) it was shown that a scaled version of this flow has asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two ‘seminal curves’ defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to 0 with increasing computational effort.

We consider a stationary face-to-face tessellation X of Rd and introduce several percolation models by colouring some of the faces black in a consistent way. Our main model is cell percolation, where cells are declared black with probability p and white otherwise. We are interested in geometric properties of the union Z of black faces. Under natural integrability assumptions, we first express asymptotic mean values of intrinsic volumes in terms of Palm expectations associated with the faces. In the second part of the paper we focus on cell percolation on normal tessellations and study asymptotic covariances of intrinsic volumes of Z ∩ W, where the observation window W is assumed to be a convex body. Special emphasis is given to the planar case where the formulae become more explicit, though we need to assume the existence of suitable asymptotic covariances of the face processes of X. We check these assumptions in the important special case of a Poisson-Voronoi tessellation.

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.