TAILIEUCHUNG - Independent And Stationary Sequences Of Random Variables - Chapter 1

Chapter 1 PROBABILITY DISTRIBUTIONS ON THE REAL LINE INFINITELY DIVISIBLE LAWS This chapter is of an introductory nature, its purpose being to indicate some concepts and results from the theory of probability which are used in later chapters . | Chapter 1 PROBABILITY DISTRIBUTIONS ON THE REAL LINE INFINITELY DIVISIBLE LAWS This chapter is of an introductory nature its purpose being to indicate some concepts and results from the theory of probability which are used in later chapters. Most of these are contained in Chapters 1-9 of Gnedenko 47 and will therefore be cited without proof. The first section is somewhat isolated and contains a series of results from the foundations of the theory of probability. A detailed account may be found in 76 or in Chapter I of 31 . Some of these will not be needed in the first part of the book in which attention is confined to independent random variables. 1. Probability spaces conditional probabilities and expectations A probability space is a triple 2 P where Q is a set of elements co 5 a cr-algebra of subsets of Q called events and P a measure on 5 with P Q 1. For E e P is called the probability of the event E. A random variable X is a real-valued measurable function on 2 5 and the measure F defined on the Borel sets of the real line R by F A P X e A is called the distribution of X. Several random variables Xr X2 . Xn may be combined in a random vector X Xr X2 . Xn and the measure F A P Xe A defined on the Borel sets of R is the distribution of X or the joint distribution of the variables Xt X2 X . More generally if T is any set of real numbers a family of random variables X t teT defined on 2 P is called a random process. Conditions for the existence of random processes with prescribed joint distributions are given by Kolmogorov s theorem 76 . A probability space is a special case of a measurable space and it is there 18 PROBABILITY DISTRIBUTIONS ON THE REAL LINE Chap. 1 fore possible to construct in it a Lebesgue integral as for example in 105 . If the function X is integrable with respect to P that is if i X co P dco oo J n then the integral f X co P dco XdP J Q Ji is called the expectation of X and is denoted by the symbol E X . If X is a random vector with values in

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