Heads or Tails - An Introduction to Limit Theorems in Probability
This book is an excellent guide to the most important limit theorems in probability for anyone who is able to follow a basic calculus course. The author uses as his “invitation to probability” a simple model of coin tossing to show the beauty and power of limit theorems in probability. There is no need to have any knowledge about measure or probability theory before reading the book. Only elementary calculus is used for proofs, making the book accessible to anyone interested in the essentials of the probability asymptotic.
The text is divided into 14 chapters. Chapters 1 to 4 contain preliminary definitions and results of discrete probability related to random variables, independence and the binomial distribution. Each of the next chapters contains one well known limit theorem stated for the sequence of independent random variables with the relevant distribution. Besides the weak law of large numbers and the central limit theorem (results that are widely known and used), more advanced theory is represented by the large and moderate deviations, the law of the iterated logarithm and the arcsine law. The results are illustrated with examples and applications in statistics are discussed to show both the utility and beauty of probability theory. The book concludes with the recurrence of random walks and the last chapter is an overview of the generalization of the results to more general random variables. The book can be recommended for students of any area of science or engineering who want to learn about the limit laws of probability. It can be also recommended to teachers as an inspiration for undergraduate introductory courses of probability or for researchers who want to have a reference book showing the strength and wide range of limit theorems in probability.