Modern Stochastics: Theory and Applications

In this paper, we consider a modified version of a well-known submartingale condition for the weak convergence of probability measures, adapted to the semi-Markov case. In this setting, it is convenient to work with an embedded Markov chain and the filtration generated by jump times. We demonstrate that a straightforward restatement of the classical result is not valid, and that an additional condition is required.

Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of $\check{\text{C}}$ech complexes of a family of sets with various sizes, growths, and shapes. The law of large numbers for persistence diagrams is established as the size of the convex window observing a marked point process tends to infinity.

In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon–Walsh construction in the Skorokhod embedding problem.

We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function systems with probabilities which are non-expansive on average.