Gaussian, Markov and stationary processes
Alejandro Ribeiro Dept. of Electrical and Systems Engineering University of Pennsylvania [email protected] http://www.seas.upenn.edu/users/~aribeiro/
November 11, 2016
Stoch. Systems Analysis Queues 1 Introduction and roadmap
Introduction and roadmap
Gaussian processes
Brownian motion and its variants
White Gaussian noise
Stoch. Systems Analysis Queues 2 Stochastic processes
I Assign a function X (t) to a random event
I Without restrictions, there is little to say about stochastic processes
I Memoryless property makes matters simpler and is not too restrictive
I Have also restricted attention to discrete time and/or discrete space
I Simplifies matters further but might be too restrictive
I Time t and range of X (t) values continuous
I Time and/or state may be discrete as particular cases
I Restrict attention to (any type or a combination of types) Markov processes (memoryless) ) Gaussian processes (Gaussian probability distributions) ) Stationary processes (“limit distributions”) )
Stoch. Systems Analysis Queues 3 Markov processes
I X (t) is a Markov process when the future is independent of the past
I For all t > s and arbitrary values x(t), x(s)andx(u)forallu < s
P X (t) x(t) X (s) > x(s), X (u) > x(u), u < s ⇥ =P X (t) x(t) ⇤X (s) > x(s) ⇥ ⇤ I Memoryless property defined in terms of cdfs not pmfs
I Memoryless property useful for same reasons of discrete time/state
I But not as much useful as in discrete time /state
Stoch. Systems Analysis Queues 4 Gaussian processes
I X (t)isaGaussianprocesswhenall prob. distributions are Gaussian
I For arbitrary times t1, t2,...tn it holds Values X (t ), X (t ),...X (t ) are jointly Gaussian ) 1 2 n I Will define more precisely later on
I Simplifies study because Gaussian distribution is simplest possible Su ces to know mean, variances and (cross-)covariances ) Linear transformation of independent Gaussians is Gaussian ) Linear transformation of jointly Gaussians is Gaussian ) I More details later
Stoch. Systems Analysis Queues 5 Markov processes + Gaussian processes
I Markov (memoryless) and Gaussian properties are di↵erent Will study cases when both hold )
I Brownian motion, also known as Wiener process
I Brownian motion with drift
I White noise linear evolution models )
I Geometric brownian motion pricing of stocks, arbitrages, risk neutral measures, pricing of stock) options (Black-Scholes)
Stoch. Systems Analysis Queues 6 Stationary processes
I Process X (t) is stationary if all probabilities are invariant to time shifts
I I.e., for arbitrary times t1, t2,...,tn and arbitrary time shift s
P[X (t + s) x , X (t + s) x ,...,X (t + s) x ]= 1 1 2 2 n n P[X (t ) x , X (t ) x ,...,X (t ) x ] 1 1 2 2 n n
I System’s behavior is independent of time origin
I Follows from our success on studying limit probabilities
I Stationary process study of limit distribution ⇡
I Will study Spectral analysis of stationary stochastic processes ) Linear filtering of stationary stochastic processes )
Stoch. Systems Analysis Queues 7 Gaussian processes
Introduction and roadmap
Gaussian processes
Brownian motion and its variants
White Gaussian noise
Stoch. Systems Analysis Queues 8 Jointly gaussian variables
I Random variables (RV) X1, X2,...,Xn are jointly Gaussian (normal) if any linear combination of them is Gaussian T I We may also say, vector RV X =[X1,...,Xn] is Gaussian (normal) T I Formally, for any a1, a2,...,an variable (a =[a1,...,an] )
T Y = a1X1 + a2X2 + ...+ anXn = a X
I is normally distributed
I Consider 2 dimensions 2RVsX and X jointly normal ) 1 2 I To describe joint distribution have to specify
Means: µ1 = E [X1]andµ2 = E [X2] ) 2 2 2 Variances: =var[X1] = E (X1 µ1) and =var[X2] ) 11 22 2 Covariance: = cov(X1)=E [(X1 µ1)(X2 µ2)] ) 12 ⇥ ⇤
Stoch. Systems Analysis Queues 9 Pdf of jointly normal RVs in 2 dimensions
T I In 2 dimensions, define vector µ =[µ1,µ2] T I And covariance matrix C with elements (C is symmetric, C = C )