Unspecified distribution in single disorder problem

We register a stochastic sequence affected by one disorder. Monitoring of the sequence is made in the circumstances when not full information about distributions before and after the change is available. The initial problem of disorder detection is transformed to optimal stopping of observed sequence. Formula for optimal decision functions is derived.Comment: 23 page

Local environment and valence state of iron in microinclusions in fibrous diamonds: X-ray Absorption and Mössbauer data

Iron valence state and local environment in a set of fibrous diamonds from Brazilian and Zairean placers were investigated using X-ray Absorption and Mössbauer spectroscopies. It is shown that the diamonds could be divided into two main groups, differing in the type of dominant Fe-bearing inclusions. In the first group Fe is mostly trivalent and is present in octahedral coordination; diamonds from the second group contain a mixture of Fe^2+^ and Fe^3+^, most likely with Fe^2+^ in dodecahedral coordination. A few other diamonds contain iron in a more reduced state: the presence of metallic Fe and Fe~3~O~4~ is inferred from XAS measurements. Spatially resolved XANES and Mössbauer measurements on polished diamond plates show that in some cases the Fe valence state may change considerably between the core and rim, whereas in other cases Fe speciation and valence remain constant. It is shown that Fe valence does not correlate with water and/or carbonate content or ratio, suggesting that iron is a minor element in the growth medium of fibrous diamonds and plays a passive role. This study suggests that, when present, evolution of the C isotopic composition with diamond growth is largely due to changes in chemistry of the growth medium and not due to variations of fO~2~

On an Effective Solution of the Optimal Stopping Problem for Random Walks

We find a solution of the optimal stopping problem for the case when a reward function is an integer function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, ? , T} converges with an exponential rate as T approaches infinity to the limit under the assumption that jumps of the random walk are exponentially bounded.optimal stopping; random walk; rate of convergence; Appell polynomials

Translational tilings by a polytope, with multiplicity

We study the problem of covering R^d by overlapping translates of a convex body P, such that almost every point of R^d is covered exactly k times. Such a covering of Euclidean space by translations is called a k-tiling. The investigation of tilings (i.e. 1-tilings in this context) by translations began with the work of Fedorov and Minkowski. Here we extend the investigations of Minkowski to k-tilings by proving that if a convex body k-tiles R^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski's conditions for 1-tiling polytopes. Conversely, in the case that P is a rational polytope, we also prove that if P is centrally symmetric and has centrally symmetric facets, then P must k-tile R^d for some positive integer k

A quickest detection problem with an observation cost

In the classical quickest detection problem, one must detect as quickly as possible when a Brownian motion without drift "changes" into a Brownian motion with positive drift. The change occurs at an unknown "disorder" time with exponential distribution. There is a penalty for declaring too early that the change has occurred, and a cost for late detection proportional to the time between occurrence of the change and the time when the change is declared. Here, we consider the case where there is also a cost for observing the process. This stochastic control problem can be formulated using either the notion of strong solution or of weak solution of the s.d.e. that defines the observation process. We show that the value function is the same in both cases, even though no optimal strategy exists in the strong formulation. We determine the optimal strategy in the weak formulation and show, using a form of the "principle of smooth fit" and under natural hypotheses on the parameters of the problem, that the optimal strategy takes the form of a two-threshold policy: observe only when the posterior probability that the change has already occurred, given the observations, is larger than a threshold $A\geq0$, and declare that the disorder time has occurred when this posterior probability exceeds a threshold $B\geq A$. The constants $A$ and $B$ are determined explicitly from the parameters of the problem.Comment: Published at http://dx.doi.org/10.1214/14-AAP1028 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org