Probability Spaces

Sample Space (any nonempty set), Probability Spaces Set of Events a sigma-algebra over (closed under complementation and countable unions) A probability space is a triple Probability Measure (a countably additive function such that http://en.wikipedia.org/wiki/Probability_space

probability space is a function Random Variables that is measureable ( for every open ) The expectation A (complex valued) random variable on a The distribution of a random variable is the measure on that satisfies for every open ) Problem 1. Show that where is the identity function.

The identity function is a Examples random variable that models the number of a randomly thrown dice. The identity function is a real random variable whose distribution is Gaussian with mean and variance Problem 2. Compute and

is the probability space where: is the sigma-algebra over Products generated by the sets in and is the unique countably additive satisfies whose restriction to The product of probability spaces Andrei NikolajevichKolmogorov (1950) The modern measure-theoretic foundation of probability theory; the original German version (GrundbegriffederWahrscheinlichkeitrechnung) appeared in 1933 (countable products also exist)

on a probability space are independent if for all Independence Example If is the prod. of prob. spaces denotes and Random variables coordinate projections, and are random variables, then the random variables are independent.

Let The relation Correlation is an equivalence relation and denotes the set of equivalence classes of random variables satisfying be random variables on a prob. space Henceforth we identify random variables with their equivalence class. The correlation of with is denoted by It gives a scalar product and therefore a Hilbert space structure on

Problem 3. Show that if are independent then Correlation Properties Problem 4. Show that if then Henceforth we identify random variables with their equivalence class.

The Gramm matrix for is Correlation Properties Thm are linearly dependent iff Proof Let and define Then

A (discrete) random process is a sequence of on a prob. space random variables Random Processes The process is wide sense stationary if and the correlations (they depend only on i-j) Problem 5. Prove that the correlation sequence is positive definite.

Henceforth we consider a wide sense stationary random process with Spectral Measure correlation sequence Herglotz’s theorem implies that there exists a measure on the circle group such that This is the spectral measure of the process. It encodes all of the correlation properties of the processes such as predictability.

Theorem 3.1.1 There exists a unique isometry Spectral Process such that Proof For any trigonometric polynomial So the result follows since trigonometric polynomials are dense in and hence dense in

Definition 3.1.1 A process is degenerate if is finite dimensional. Problem 6 Show this holds iff Denegerate Processes where Problem 7 Show that if and are distinct points in then is a basis for Suggestion Show that is isomorphic to with scalar product Then use the fundamental theorem of algebra to show are lin. ind. in

Lemma 3.1.1 A measure has the form iff Denegerate Processes Proof The n-th column of the matrix is where so rank of matrix is < N+1 so det = 0. If det = 0 then since the matrix is a Gram matrix for the Hilbert space are linearly dependent. Problem 8. Show this implies

Definition Define by Denegerate Processes Problem 9. Show that is unitary. Continuation of Proof. Assuming that is unitary and there exists an orthonormal basis for such that If then so

White Noise is a stationary random process with and correlation sequence Examples of Stationary Processes A stationary random process is white noise iff its spectral measure equals If then given by is a stationary random process. Problem 10. Compute the spectral measure of

Probability Spaces