“Ergodic process”的意思、由来-开放百科全书

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Ergodic process

释义

Specific definitions
Discrete-time random processes
Examples of non-ergodic random processes
See also
Notes
References

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.^[1]

Specific definitions

One can discuss the ergodicity of various statistics of a stochastic process. For example, a wide-sense stationary process has constant mean

,

and autocovariance

,

that depends only on the lag and not on time . The properties and

are ensemble averages not time averages.

The process is said to be mean-ergodic^[2] or mean-square ergodic in the first moment^[3]

if the time average estimate

converges in squared mean to the ensemble average as .

Likewise,

the process is said to be autocovariance-ergodic or d moment^[3]

if the time average estimate

converges in squared mean to the ensemble average , as .

A process which is ergodic in the mean and autocovariance is sometimes called ergodic in the wide sense.

Discrete-time random processes

The notion of ergodicity also applies to discrete-time random processes

for integer .

A discrete-time random process is ergodic in mean if

converges in squared mean

to the ensemble average ,

as .

Examples of non-ergodic random processes

An unbiased random walk [https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk] is non-ergodic. Its expectation value is zero at all times, whereas its time average is a random variable with divergent variance.

Suppose that we have two coins: one coin is fair and the other has two heads. We choose (at random) one of the coins first, and then perform a sequence of independent tosses of our selected coin. Let X[n] denote the outcome of the nth toss, with 1 for heads and 0 for tails. Then the ensemble average is ½ (½ + 1) = ¾; yet the long-term average is ½ for the fair coin and 1 for the two-headed coin. So the long term time-average is either 1/2 or 1. Hence, this random process is not ergodic in mean.

Notes

1. ^Originally due to L. Boltzmann. See part 2 of {{cite book|title=Vorlesungen über Gastheorie|year= 1898 |location=Leipzig|publisher=J. A. Barth|url=https://books.google.com/books?id=99IEAAAAYAAJ&oe=UTF-8 |oclc=01712811}} ('Ergoden' on p.89 in the 1923 reprint.) It was used to prove equipartition of energy in the kinetic theory of gases
2. ^Papoulis, p.428
3. ^¹Porat, p.14

References

{{cite book

| last = Porat
| first = B.
| title = Digital Processing of Random Signals: Theory & Methods
| date = 1994
| publisher = Prentice Hall
| isbn = 0-13-063751-3
| pages = 14 }}

{{cite book

|author=Papoulis, Athanasios
|title=Probability, random variables, and stochastic processes
|publisher=McGraw-Hill
|location=New York
|year=1991
|pages=427–442
|isbn=0-07-048477-5
|oclc=
|doi=
|accessdate=
}}

2 : Ergodic theory|Signal processing

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Specific definitions

Discrete-time random processes

Examples of non-ergodic random processes

See also

Notes

References