A conditional limit theorem for high-dimensional ℓᵖ-spheres

The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

Multi-type subcritical branching processes in a random environment

We study the asymptotic behavior of the survival probability of a multi-type branching process in a random environment. In the one-dimensional situation, the class of processes considered corresponds to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.

The longitudinal-transverse bending of reinforced concrete rods

The longitudinal transverse bending of multilayer rods made of reinforced concrete of arbitrary cross-section is considered. It is assumed that the coefficients in the connection equation between stresses, strains and temperature are different in each layer. The achievement of maximum deformation of the permissible limit value at stretching or compression is accepted as the criterion of conditional limit state in the i-layer. The case of longitudinal-transverse bending of a hinged rod is considered as an example of this method of solution. The distribution of bending moments and longitudinal forces, displacements and deformations is determined.

Efficiency, Sufficiency, and Optimality

In this chapter I provide additional rationalization for using the info-metrics framework. This time the justifications are in terms of the statistical, mathematical, and information-theoretic properties of the formalism. Specifically, in this chapter I discuss optimality, statistical and computational efficiency, sufficiency, the concentration theorem, the conditional limit theorem, and the concept of information compression. These properties, together with the other properties and measures developed in earlier chapters, provide logical, mathematical, and statistical justifications for employing the info-metrics framework.

Compression strength perpendicular to grain of Serbian spruce (Picea omorika (Pancic) purkyně) wood from plantations and natural stands

This paper presents the results of testing the compression of Serbian spruce wood from plantations and natural stands. Compression perpendicular to grain in radial and tangential direction was tested. A dilatation of 1% was taken for a conditional boundary dilatation, and the appropriate strength for the conditional limit strength was taken. Six trees from plantations and nine trees from natural stands were analyzed. In total, 309 samples were tested. The regression analysis examined the dependence of these mechanical properties on the width of the annual rings, the percentage of late wood and wood density.

Weakening the independence assumption on polar components: limit theorems for generalized elliptical distributions

By considering the extreme behavior of bivariate random vectors with a polar representation R(u(T), v(T)), it is commonly assumed that the radial component R and the angular component T are stochastically independent. We investigate how to relax this rigid independence assumption such that conditional limit theorems can still be deduced. For this purpose, we introduce a novel measure for the dependence structure and present convenient criteria for validity of limit theorems possessing a geometrical meaning. Thus, our results verify a stability of the available limit results, which is essential in applications where the independence of the polar components is not necessarily present or exactly fulfilled.

Vulnerability Assessment of Damaged Classical Multidrum Columns

Multi-drum columns are articulated structures, made of several discrete bulgy stone blocks (drums) placed one on top of the other without mortar. The multi-drum column is a typical structural element of temples of the Classical, Hellenistic and earlier Roman period. Despite the lack of any lateral load resisting mechanism, these columns have survived several strong earthquakes over the centuries. The Chapter focuses on the effect of past drum dislocations on the vulnerability of classical columns and presents a performance-based framework for their seismic risk assessment. The vulnerability is numerically calculated through response estimations using detailed three-dimensional models based on the Discrete Element Method. Conditional limit-state probabilities are calculated and appropriate performance criteria are suggested. The proposed methodology is able to pinpoint cases where past damage affects the vulnerability of such structures and can serve as a valuable decision-making tool.