A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence variables [Formula: see text] drawn independently from same distribution, goal to choose time τ such that maximum value α all distributions, text]. What makes this problem challenging decision whether may only depend on value...

Abstract We discuss estimating the probability that sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $$\mathbb {P}(\sum _{i=1}^{N}{X_i} \le \gamma )$$ P ( ∑ i = <mml...

Suppose a large economy with individual risk is modeled by a continuum of pairwise exchangeable random variables (i.i.d., in particular). Then the relevant stochastic process is jointly measurable only in degenerate cases. Yet in Monte Carlo simulation, the average of a large finite draw of the random variables converges almost surely. Several necessary and sufficient conditions for such “Monte...

We provide a necessary and sufficient condition for the ratio of two jointly α-Fréchet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As...

Information bottleneck [IB] is a technique for extracting information in some ‘input’ random variable that is relevant for predicting some different ‘output’ random variable. IB works by encoding the input in a compressed ‘bottleneck variable’ from which the output can then be accurately decoded. IB can be difficult to compute in practice, and has been mainly developed for two limited cases: (1...

Nonlinear transformation is one of the major obstacles to analyzing the properties of multilayer perceptrons. In this letter, we prove that the correlation coefficient between two jointly Gaussian random variables decreases when each of them is transformed under continuous nonlinear transformations, which can be approximated by piecewise linear functions. When the inputs or the weights of a mul...

Fuzzy random variable is a measurable function from a probability space to the set of fuzzy variables, while random fuzzy variable is a function from a credibility space to the set of random variables. The concepts of independent and identically distributed fuzzy random variables and random fuzzy variables are presented, and some useful properties are discussed.