Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."

Measure Thoery: Chp 10. Probability

10.1 Basics

A probability space is a measure space such that . The elements of are called the elementary outcomes or the sample points of our experiment, and the members of are called events. If , then is the probability of the event A.

A real-valued random variable on a probability space is an -measurable function from to . More generally, a random variable with values in a measurable space is a measurable function from to .

If a real-valued random variable on a probability space is integrable with respect to , then its expected value, or expectation, written E(X), is the integral of X with respect to P.

One calls the expected value of the variance of ; it gives a measure of the amount by which the values of differ from the expected value of .

Function defined by

The function is called the cumulative distribution function.

10.2 Laws of Large Numbers

Let and be random variables on the probability space . Then is said to converge in probability to if

holds for each positive number and to converge almost surely to (or to converge a.s. to X) if

In other words, converges to in probability if it converges to in measure, and converges to almost surely if it converges to almost everywhere.

Theorem 10.2.1 (Weak Law of Large Numbers). Let be a sequence of independent identically distributed real-valued random variables with finite second moments. For each let . Then converges to in probability.

Theorem 10.2.5 (Strong Law of Large Numbers). Let be a sequence of independent identically distributed random variables with finite expected values. For each let . Then converges to almost surely.

Proposition 10.2.6 (Kolmogorov’s Inequality). Let be independent random variables, each of which has mean and a finite second moment, and for each let . Then

holds for each positive .

Theorem 10.2.8 (Converse to the Strong Law of Large Numbers). Let be a sequence of independent identically distributed random variables that do not have finite expected values. For each let . Then almost surely.

10.3 Convergence in Distribution and the Central Limit Theorem

Theorem 10.3.16 (Central Limit Theorem). Let be a sequence of independent identically distributed random variables, with common mean and variance , and for each let . Then the normalized sequence converges in distribution to a normal (i.e., Gaussian) distribution with mean and variance .

10.4 Conditional Distributions and Maringales

Theorem 10.4.8 (Doob’s Martingale Convergence Theorem). Let be a probability space, and let be a submartingale on such that . Then the limit exists almost surely, and .

10.5 Brownian Motion

Suppose that is a probability space and that is either or . A stochastic process with values in is a Brownian motion if - , - for each choice of such that the increments , are independent, with having distribution , that is, a normal distribution with mean and variance , and - for each the function defined by is continuous.

Theorem 10.5.1. Let . Then a one-dimensional Brownian motion with parameter set exists. That is, there exist a probability space and random variables , on such that the stochastic process is a Brownian motion.

10.6 Construction of Probability Measures

Theorem 10.6.2 (Kolmogorov Consistency Theorem). Let be a nonempty set, let ${(i, i)}{i I} $ be an indexed family of measurable spaces, and let be the collection of all nonempty finite subsets of . As in the discussion above, define product measurable spaces and , plus projections and , where and . Let be an indexed family of probability measures on the spaces . If - the measurable spaces are all standard, and - the measures ${ P{I_0}}_{I_0 } $ are consistent, in the sense that they satisfy condition (2)

then there is a unique probability measure on such that for each in the distribution of is .

References

[1] @book{cohn2013measure, title={Measure theory}, author={Cohn, Donald L}, year={2013}, publisher={Springer} }