Personalizing Rules by Age

To complete the demonstration of personalized law in action, this chapter focuses on the inputs used to tailor the individualized commands. One input that is likely to feature in the personalization of many rules is age. Age is informative because it is often correlated with personal attributes that matter to achieving the goals of a law. Preferences, cognition, judgment, experience, and physical ability—all vary with age. Young age is a factor in the denial of legal capacity and in the conferral of various paternalistic protections, whereas old age represents changing needs, capacities, and entitlements. Under personalized law, age would be an input affecting legal commands that are currently age-invariant, such as the intestate succession default rule or speed limit laws. In addition, age of capacity laws, which are currently used to regulate entry into various activities, would use different age cutoffs for different people.

On Sinaĭ Billiards on Flat Surfaces with Horns

We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).

Rate of convergence in total variation for the generalized inverse Gaussian and the Kummer distributions

The generalized inverse Gaussian distribution converges in law to the inverse gamma or the gamma distribution under certain conditions on the parameters. It is the same for the Kummer’s distribution to the gamma or beta distribution. We provide explicit upper bounds for the total variation distance between such generalized inverse Gaussian distribution and its gamma or inverse gamma limit laws, on the one hand, and between Kummer’s distribution and its gamma or beta limit laws on the other hand

Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized T02 statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples.

On rates of convergence in the probabilistic analysis of algorithms

Within the last thirty years, the contraction method has become an important tool for the distributional analysis of random recursive structures. While it was mainly developed to show weak convergence, the contraction approach can additionally be used to obtain bounds on the rate of convergence in an appropriate metric. Based on ideas of the contraction method, we develop a general framework to bound rates of convergence for sequences of random variables as they mainly arise in the analysis of random trees and divide-and-conquer algorithms. The rates of convergence are bounded in the Zolotarev distances. In essence, we present three different versions of convergence theorems: a general version, an improved version for normal limit laws (providing significantly better bounds in some examples with normal limits) and a third version with a relaxed independence condition. Moreover, concrete applications are given which include parameters of random trees, quantities of stochastic geometry as well as complexity measures of recursive algorithms under either a random input or some randomization within the algorithm.

Recurrent Sequences Play for Survival Probability of Discrete Time Risk Model

In this article we investigate a homogeneous discrete time risk model with a generalized premium income rate which can be any natural number. We derive theorems and give numerical examples for finite and ultimate time survival probability calculation for the mentioned model. Our proved statements for ultimate time survival probability calculation, at some level, are similar to the previously known statements for non-homogeneous risk models, where required initial values of survival probability for some recurrent formulas are gathered by certain limit laws. We also give a simplified proof that a ruin is almost unavoidable with a neutral net profit condition and state several conjectures on a certain type of recurrent matrices non-singularity. All the research done can be interpreted as a possibility that symmetric or asymmetric random walk (r.w.) hits (or not) the line u+κt and that possibility is directly related to the expected value of r.w. generating random variable which might be equal, above or bellow κ.