Probability Spaces A probability space is a triple (closed under Sample Space (any nonempty set), Set of Events a sigma-algebra over complementation and countable unions) Probability Measure (a countably additive function such that

Random Variables A (complex valued) random variable on a that is measureable ( is a functionprobability space for every open) The expectation The distribution of a random variable is the measurethat satisfieson for every open ) Problem 1. Show that whereis the identity function.

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

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

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

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

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

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

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

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

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

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

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

Denegerate Processes Problem 9. Show that Definition Define is unitary. by Continuation of Proof. Assuming that there exists an orthonormal forsuch that is unitary and basis Ifthen so

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