Stochastic Processes 1
Definition and Classification X(, t): stochastic process: X : T! R (, t) X(, t) where is a sample space and T is time. {X(, t) is a family of r.v. defined on {, A, P and indexed by t T. For a fixed 0, X(( 0,t) is a function of time or a sample function For a fixed t 0, X((, t 0 ) is a random variable or a function of Discrete-time random process: n T, X(, n) Continuous-time random process: t T X(, n)
Probabilistic characterization of X(, t) {X(, t)} is an infinite family of r.v., we then need to know: 1. First-order p.d. or distribution. Second-order p.d. or distribution F (x, t) =P [X(, t) apple x] for all t @F(x, t) f(x, t) = @x F (x 1,x,t 1,t )=P [X(, t 1 ) apple x 1,X(, t ) apple x ] for all t 1,t T f(x 1,x,t 1,t )= @ F (x, t) @x 1 @x 3. n-th order p.d. or distribution F (x 1,,x n,t 1,,t n )=P [X(, t 1 ) apple x 1,,X(, t n ) apple x n ] for all t 1,,t n T f(x 1,,x n,t 1,,t n )= @n F (x, t) @x 1 @x n Remark The n-th order p.d./p.d must also satisfy Symmetry F (x 1,,x n,t 1,,t n )=F (x i1,,x in,t i1,,t in ) for any permutation of {1,,,n}. Consistency in marginals F (x 1,,x n 1,t 1,,t n 1 )=F (x 1,,x n 1, 1,t 1,,t n ) 3
Kolmogorov s Theorem If F (x 1,,x n,t 1,,t n ) for all sets of {t i }T satisfy the symmetry and consistency conditions, there corresponds a stochastic process. Second-order Properties F (x, t),f(x, t), first-order p.d./p.d, and F (x 1,x,t 1,t ),f(x 1,x,t 1,t ) are not su cient to characterize X(, t) but provide important information: Mean function (t) X (t) =E(X(t)) = Z 1 1 xf(x, t)dx function of t that depends on the first-order p.d. Auto-covariance function C X (t 1,t ) C X (t 1,t ) = E[(X(t 1 ) (t 1 ))(X(t ) (t ))] = E[X(t 1 )X(t )] R X (t 1,t ) (t 1 ) (t ) R X (t 1,t ) is the autocorrelation function, a function of t 1, t and secondorder probability density ZZ R X (t 1,t )=E[X(t 1 )X(t )] = x 1 x f(x 1,x,t 1,t )dx 1 dx 4
Remarks Note that C X (t, t) =E[(X(t) (t)) ]= X(t) or the variance of X(t). C X (t 1,t ) relates r.v. s at times t 1 and t,i.e.,intime,whilec X (t, t) relates the r.v. X(t) with itself, i.e., in space. Correlation coe cient r X (t 1,t )= C X(t 1,t ) X(t 1 ) X (t ) Cross-covariance X(t), Y (t) real processes C XY (t 1,t ) = E[(X(t 1 ) X (t 1 ))(Y (t ) Y (t ))] = E[X(t 1 )Y (t )] R XY (t 1,t ) X (t 1 ) Y (t ) R XY (t 1,t ) is the cross-correlation function. X(t), Y (t) are uncorrelated i C XY (t 1,t ) = 0 for all t 1 and t T. 5
Remark A zero-mean process (t) is called strictly white noise if (t i ) and (t j ) are independent for t i 6= t j. It is called white noise if it is uncorrelated for di erent values of t, i.e., C (t i,t j )= (t i t j )= t i = t j 0 t i 6= t j Gaussian Processes X(t) is Gaussian if {X(t 1 ),,X(t n ) are jointly Gaussian for any value of n and times {t 1,,t n }. Remark The statistics of a Gaussian process are completely determines by the mean (t) and the covariance C X (t 1,t ) functions. Thus f(x, t) requires (t),c(t, t) = X(t) f(x 1,x,t 1,t ) requires (t 1 ), (t ),C(t 1,t ),C(t 1,t 1 ),C(t,t ) and in general the n th -order characteristic function X(bf!, t) =exp 4j X i! i X (t i ) 0.5 X i,k 3! i! k C X (t i,t j ) 5 6
Stationarity Strict Stationarity X(t) is strictly stationary i F (x 1,,x n,t 1,,t n )=F (x 1,,x n,t 1 +,,t n + ) for all n and values.i.e., statistics do not change with time. If X(t) is strictly stationary then Proof its k th -moments are constant, i.e.,e[x k (t)] = constant for all t. R X (t, t + ) =R X (0, ), or it only depends on the lag Using strict stationarity E[X k (t)] = Z 1 1 x k f(x, t)dx = Z 1 1 x k f(x, t + )dx = E[X k (t + )] for any, which can only happen when it is a constant. By strict stationarity ZZ R X (t, t + ) = x 1 x f(x 1,x,t,t+ )dx 1 dx = ZZ x 1 x f(x 1,x, 0, )dx 1 dx = R X (0, ) Wide-sense Stationarity (wss) X(t) is wss if 1. E[X(t)] = constant, Var[X(t)] = constant, for all t.. R X (t 1,t )=R X (t t 1 ) 7
Examples of random processes Discrete-time Binomial process Consider the Bernoulli trials where X(n) = 1 event occurs at time n 0 otherwise with P [X(n) = 1] = p, P [X(n) = 0] = 1 p = q. The discrete-time Binomial process counts the number of times the event occurred (successes) in a total of n trials, or Y (n) = X(i), Y(0) = 0, n 0 Since Y (n) = 1 X(i) + X(n) = Y (n 1) + X(n) the process is also represented by the di erence equation Y (n) =Y (n 1) + X(n) Y (0) = 0, n 0 8
Characterization of Y (n) First-order p.d. f(y, n) = P [Y (n) =k] (y k) = k=0 k=0 n k (y k), 0 apple k apple n Second and higher-order p.d. are di Y (n) s. Statistical averages cult to find given the dependence of the E[Y (n)] = E[X(i)] p = np Var[Y (n)] = E[(Y (n) np) ]=E[( = X i,j = = npq (X(i) p) ] E[(X(i) p)(x(j) p)] = X i,j E[(X(i) p) ] Var(X(i))=(1 p+0 q) p =pq E[X(i) p]e[x(j) p] =0 if i6=j independence of X(i) Notice that E[Y (n)] = np, so it depends on time n, and Var[Y (n)] = npq also a function of time n, so the discrete binomial process is non-stationary. 9
Random Walk Process Let the discrete binomial process be defined by Bernoulli trials s event occurs at time n X(n) = s otherwise so that Y (n) = X(i), Y(0) = 0, n 0 Possible values at times bigger than zero n =1 Y (1) = 1, 1 n = Y () =, 0, n =3 Y (3) = 3, 1, 1, 3.. In general, at step n = n 0 Y (n) can take values {k n 0, 0 apple k apple n 0 }, so for instance for n =, Y () can take values k, 0 apple k apple or, 0,. 10
Characterization of Y (n) First-order p. mass d. P [Y (n 0 )=k n 0, 0 apple k apple n 0 ] Convert the random walk process into a binomial process by (s = 1) Z(n) = X(n)+1 1 event occurs at time n = 0 otherwise Y (n) = X(i) = (Z(i) 1) = Z(i) n =Ỹ (n) n where Ỹ (n) is the binomial process in the previous example. Letting m =k n 0, 0 apple k apple n 0 then we have for 0 apple n 0 apple n P [Y (n 0 )=m] = P [Ỹ (n 0) n 0 =k n 0 ]=P[Ỹ(n 0)=k] = Mean and variance n0 k p k q n 0 k E[Y (n)] = E[Ỹ (n) n] =E[Ỹ (n)] n =np n Var[Y (n)] = 4Var[Ỹ (n)] = 4npq Both of which depend on n, so that the process is nonstationary. 11
Sinusoidal processes X(t) =A cos( t + ) A constant,frequency and phase are random and independent. Assume U[, ]. The sinusoidal process X(t) isw.s.s. The E[X(t)] = AE[cos( t + )] =AE[cos( t) sin( ) sin( t) cos( )] = A[E[cos( t)] E[sin( )] 0 = 0 E[sin( )] = Z E[sin( t)] E[cos( )]] independence 0 1 sin( )d =0 area under a period of the sinusoid T 0 =, likewise for E[cos( )] = 0. The autocorrelation R X (t +,t) = E[X(t + )X(t)] = A E[cos( (t + ) + ) cos( t + )] = A E[cos( ) + cos( t + + )] A E[cos( t + ) sin( )] A E[sin( t + ) cos( )] 0 = A E[cos( )] + = A E[cos( )] where the zero term is obtained because of the independence and that the expected values E[sin( )] = E[cos( )] = 0 by similar reasons as above. 1
X(t) =A cos(! 0 t), A U[0, 1],! 0 constant. X(t) is non-stationary. t X(t) 0 A cos(0) = A U[0, 1] 4! 0 A cos( /4 ) = Ap U[0, p /]! o A U[ 1, 0] For each time the first-order p.d. is di erent so the process is not strictly stationary. The process can be shown not to be wide-sense stationary: E[X(t)] = cos(! 0 t)e[a] =0.5 cos(! 0 t) R X (t +,t) = E[A ] cos(! 0 (t + )) cos(! 0 t) = A[cos(! 0 ) + cos(! 0 (t + ))] R X (t, t) = Var[X(t)] = A(1 + cos(! 0 t) which gives that the mean and the variance are not constant, and the autocorrelation is not a function of the lag, therefore the systems is not wide-sense stationary. 13
Gaussian processes Let X(t) =A + Bt, wherea and B are jointly Gaussian. Determine if X(t) is Gaussian. Consider 6 4 X(t 1 ). X(t n ) 3 7 5 = 6 4 1 t 1.. 1 t n 3 7 5 apple A B for any n and times {t k }, 1 apple k apple n, the above is a linear combination of A and B which are Gaussian then {X(t k )} are jointly Gaussian, and so the process X(t) is Gaussian. Gaussian processes Consider a Gaussian process W (n), 1 < n < 1 such that E[W (n)] = 0 for all n R(k, `) = (k `) = k = ` 0 k 6= ` such a process is a discrete white noise, determine its n th -order p.d. The covariance is C(k, `) =R(k, `) E[W (k)]e[w (`)] = R(k, `) which is zero when k 6= `, sow (k) and W (`) are uncorrelated and by being Gaussian they are independent. So f(w n1,,w nm )= Y i f(w ni ) 14