in this paper, we extend some famous maximal inequalities and obtain strong laws of largenumbers for arbitrary random variables by use of these inequalities and martingale techniques.

Many of the methods in computational statistics require the ability to generate random variables from known probability distributions. This is at the heart of Monte Carlo simulation for statistical inference (Chapter 6), bootstrap and resampling methods (Chapters 6 and 7), Markov chain Monte Carlo techniques (Chapter 11), and the analysis of spatial point processes (Chapter 12). In addition, we...

In this paper we initiate the study of discrete random variables over domains. Our work is inspired by work of Daniele Varacca, who devised indexed valuations as models of probabilistic computation within domain theory. Our approach relies on new results about commutative monoids defined on domains that also allow actions of the non-negative reals. Using our approach, we define two such familie...

We extend Fano’s inequality, which controls the average probability of (disjoint) events in terms of the average of some Kullback-Leibler divergences, to work with arbitrary [0, 1]–valued random variables. Our simple two-step methodology is general enough to cover the case of an arbitrary (possibly continuously infinite) family of distributions as well as [0, 1]–valued random variables not nece...

Stochastic models for phenomena that can exhibit sudden changes involve the use of processes whose sample functions may have discontinuities. This report provides some tools for working with such processes. We develop a sample path formula for the cumulative jump height over a given time interval. From this formula an expression for the expected value of the cumulative jump random variable is d...

Contents 1. Introduction. 2. The distribution function of a random variable. 3. Classification of random variables. 4. Some special discrete probability distributions. 5. Some special continuous probability distributions. 6. Location characteristics of a real-valued random variable. 7. Dispersion characteristics of a real-valued random variable. 8. Joint distribution functions. 9. Independence ...

For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, we have very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomial-time algorithms for these problems fail to find solutions even at much lower densities. To understand the origin of this gap one can study how the structure...

For a particular decision situation, the analyst must assign a distribution to each random variable. One method is to perform repeated replications of the experiment. Statistical analysis provides estimates of the probability of each possible occurrence. Another, and often more practical method, is to identify the distribution to be one of the named distributions. It is much easier to estimate ...

Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes them from discrete random variables, which can take on only a sequence of values, usually integers. Typically random variables that represent, for example, time or distance will be continuous rather than discrete. Just as we descri...