An Introduction to Probability Theory and its Applications, Volume I

TL;DR: In this paper, the authors considered a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference: if there are k individuals in the n th generation, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case.

TL;DR: In this paper, the authors propose an alternative approach to synthesize polynomial loop invariants by using Stengle's Positivstellensatz and a transformation to a sum-of-squares problem.

TL;DR: It is shown that what is called the generalized ergodicity coefficient, which is the group generalized inverse of A = I − T, is the smallest condition number of Markov chains with respect to the (p, ∞)-norm pair, and identified κ3 and κ6 in the Cho–Meyer list of 8 condition numbers.

TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.

TL;DR: This work addresses the problem of predicting a word from previous words in a sample of text and discusses n-gram models based on classes of words, finding that these models are able to extract classes that have the flavor of either syntactically based groupings or semanticallybased groupings, depending on the nature of the underlying statistics.

TL;DR: In this article, the authors present a case study of non-normal distribution and non-commutative joint distributions and define a set of basic combinatorics, such as non-crossing partitions, sum-of-free random variables, and products of free random variables.

TL;DR: This chapter explains why many real-world networks are small worlds and have large fluctuations in their degrees, and why Probability theory offers a highly effective way to deal with the complexity of networks, and leads us to consider random graphs.