A two-step sequential procedure for detecting an epidemic change

The typical approach in change-point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, and this is our approach, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. In this paper we focus on epidemic changes, that is, a first change (the outbreak) when there is a change in the distribution, and a second change, when the process regains its ordinary structure. Based on the counting process related to the original process observed at equidistant time points, we propose some stopping rules for this to happen and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes, extreme value asymptotics for Gaussian processes, and the law of the iterated logarithm.     

Antiangiogenic Activity and in Silico Cereblon Binding Analysis of Novel Thalidomide Analogs

Due to its antiangiogenic and anti-immunomodulatory activity, thalidomide continues to be of clinical interest despite its teratogenic actions, and efforts to synthesize safer, clinically active thalidomide analogs are continually underway. In this study, a cohort of 27 chemically diverse thalidomide analogs was evaluated for antiangiogenic activity in an ex vivo rat aorta ring assay. The protein cereblon has been identified as the target for thalidomide, and in silico pharmacophore analysis and molecular docking with a crystal structure of human cereblon were used to investigate the cereblon binding abilities of the thalidomide analogs. The results suggest that not all antiangiogenic thalidomide analogs can bind cereblon, and multiple targets and mechanisms of action may be involved.     

Precise asymptotics – A general approach

The legendary 1947-paper by Hsu and Robbins, in which the authors introduced the concept of “complete convergence”, generated a series of papers culminating in the like-wise famous Baum–Katz 1965-theorem, which provided necessary and sufficient conditions for the convergence of the series $\sum_{n=1}^{\infty}n^{r/p-2}P (|S_{n}| \geqq \varepsilon n^{1/p})$ for suitable values of r and p, in which S _n denotes the n-th partial sum of an i.i.d. sequence. Heyde followed up the topic in his 1975-paper where he investigated the rate at which such sums tend to infinity as ε↘0 (for the case r=2 and p=1). The remaining cases have been taken care later under the heading “precise asymptotics”. An abundance of papers have since then appeared with various extensions and modifications of the i.i.d.-setting. The aim of the present paper is to show that the basis for the proof is essentially the same throughout, and to collect a number of examples. We close by mentioning that Klesov, in 1994, initiated work on rates in the sense that he determined the rate, as ε↘0, at which the discrepancy between such sums and their “Baum–Katz limit” converges to a nontrivial quantity for Heyde’s theorem. His result has recently been extended to the complete set of r- and p-values by the present authors.     

Limit Theorems for Maxima of Sums and Renewal Processes

Let {X_i}_i=1,2,... be a sequence of i.i.d. random variables, let S_n = X₁ + ... + X_n, and let S_n a.s. We discuss necessary and sufficient conditions for the Kolmogorov and Marcinkiewicz–Zygmund type strong laws of large numbers and for the law of the iterated logarithm for renewal processes defined in two different ways. Bibliography: 16 titles.     

Self-Normalized LIL for Hanson–Russo Type Increments

Let X ₁, X ₂,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given.     

Invariance principles for renewal processes when only moments of low order exist

Starting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419–1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived.     

Asymptotics for increments of stopped renewal processes

Motivated by our earlier work on change-point analysis we prove a number of limit theorems for increments of renewal counting processes, or the corresponding first passage times. The starting point of the increments is deterministic as well as random, a typical example being the first stopping time to detect a change-point of some (continuously) observed process.     

On factorization representations for Avakumović--Karamata functions with nondegenerate groups of regular points

Avakumovi-Karamata functions f are generalized regularly varying functions (so--called ORV functions) such that f*()= limsup _x f(x)/f(x) is finite for all >0. In this paper, we investigate classes of ORV functions with "nondegenerate groups of regular points", that is, having points 1, for which f*() exists as a positive and finite limit (instead of limsup) on a nontrivial subgroup of the positive real axis. Certain factorization representations, characterizations and uniform convergence theorems are proved, describing both the structure of ORV functions f as well as that of their limit functions f*. Some well-known results from regular variation theory are covered by this general approach.