Ergodic process

In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. The reasoning is that any collection of random samples from a process must represent the average statistical properties of the entire process. In other words, regardless of what the individual samples are, a birds-eye view of the collection of samples must represent the whole process. Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent rate. In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. The reasoning is that any collection of random samples from a process must represent the average statistical properties of the entire process. In other words, regardless of what the individual samples are, a birds-eye view of the collection of samples must represent the whole process. Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent rate. One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process X ( t ) {displaystyle X(t)} has constant mean and autocovariance that depends only on the lag τ {displaystyle au } and not on time t {displaystyle t} . The properties μ X {displaystyle mu _{X}} and r X ( τ ) {displaystyle r_{X}( au )} are ensemble averages not time averages. The process X ( t ) {displaystyle X(t)} is said to be mean-ergodic or mean-square ergodic in the first momentif the time average estimate converges in squared mean to the ensemble average μ X {displaystyle mu _{X}} as T → ∞ {displaystyle T ightarrow infty } . Likewise,the process is said to be autocovariance-ergodic or d moment if the time average estimate converges in squared mean to the ensemble average r X ( τ ) {displaystyle r_{X}( au )} , as T → ∞ {displaystyle T ightarrow infty } .A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense. The notion of ergodicity also applies to discrete-time random processes X [ n ] {displaystyle X} for integer n {displaystyle n} .