Random process

A stochastic process (probabilistic process, random function, stochastic process) in the theory of probability is a family of random variables indexed by some parameter, most often playing the role of time or coordinates.
Other definition:
Random process is u (t), the instantaneous values ​​of which are random variables.

Definition

Let probabilistic space be given . Parameterized family random variables

,

Where An arbitrary set is called a random function.

Terminology

• If a then parameter can be interpreted as time. Then a random function called a random process. If many discretely for example then such a random process is called a random succession .

• If a where then parameter can be interpreted as a point in space, and then the random function is called a random field .

This classification is not strict. In particular, the term “random process” is often used as an unconditional synonym for the term “random function”.

Classification

• Random process called a process discrete in time , if the system in which it flows, changes its state only at times whose number is finite or countable. A random process is called a process with continuous time , if the transition from state to state can occur at any time.
• A random process is called a process with continuous states if the value of the random process is a continuous random variable. A random process is called a random process with discrete states , if the value of the random process is a discrete random variable:
• A random process is called stationary if all multidimensional distribution laws depend only on the relative position of the moments of time. but not from the actual values ​​of these quantities. In other words, a random process is called stationary if its probabilistic laws are constant in time. Otherwise, it is called non-stationary .
• A random function is called stationary in a broad sense if its expectation and variance are constant, and the ACF depends only on the difference in time points for which the ordinates of the random function are taken. The concept introduced A. Ya. Khinchin.
• A random process is called a process with stationary increments of a certain order if the probability laws of such an increment are constant over time. Such processes were considered by Yaglom ^[1] .
• If the ordinates of a random function obey the normal distribution law, then the function itself is called normal .
• Random functions, the distribution of the ordinates of which at a future time instant is completely determined by the value of the process’s ordinates at the current time and does not depend on the values ​​of the process’s ordinates at previous times, are called Markov functions.
• A random process is called a process with independent increments , if for any set where , but random variables , , , independent in the aggregate.
• If, in determining the moment functions of a stationary random process, the operation of averaging over a statistical ensemble can be replaced by averaging over time, then such a stationary random process is called ergodic .
• Among the random processes emit pulsed random processes.

Random process trajectory

Let a random process be given. . Then for each fixed - a random variable called a section . If fixed elementary outcome then - deterministic parameter function . Such a function is called a trajectory or a realization of a random function. .

Examples

• where is called the standard Gaussian (normal) random sequence.
• Let be and - random value. Then

is a random process.

Notes

See also

• Random value
• Markov chain
• Markov process
• Non-Markov process

A function X (t) is called random if its value for any argument t is a random variable. Random functions of time are called random processes.

The implementation of the random function X (t) (sampling function) is a specific form that it takes as a result of experience. The implementation of a random process can be considered as an element of the set of possible physical realizations of a random process (Fig. 5.8). The set of realizations of a random process is called an ensemble of realizations. The set of realizations at a fixed point in time (a sample of random values) is called a cross section of a random process.

Fig. 5.8. Implementing a random process

In any section, a random process is a random variable.

The expectation of a random process is a function of time.

(5.10)

The second central moment for two sections of a random process is called the covariance function

(5.11)

Where - centered random process.

At t = t ′, the covariance function is equal to the variance of the random process.

(5.12)

The mathematical expectation and the covariance function of a random process can be found by the realizations of the random process — by the averages by realizations:

(5.13)

where N is the number of realizations of the random process.

If the expectation and covariance function are independent of time t, then the process is stationary:

(5.14)

where τ = t ′ - t. In (5.14), the covariance function depends only on the magnitude of τ, and not on its location on the time axis (Fig. 5.9).

Fig. 5.9. The time between two sections of a random process

A possible view of the covariance function is shown in Fig. 5.10.

In many cases, the normalized covariance (or correlation) function is used. For stationary random process

(5.15)

The magnitude of the correlation function | r _X (τ) | ≤ 1.

The time averaging of individual sampling functions (implementations) is possible. For the k-th sampling function, we have:

(5.16)

Fig.5.10. Covariance function of a random process

If the random process X (t) is stationary and the characteristics m _X and R _X (τ) are the same for different sampling functions, then this process is called ergodic.

Ergodic processes are an important class of random processes.

Non-stationary random processes are all random processes that do not have the properties of stationarity. These processes are complex in research, and often in analysis problems they are divided into stationarity intervals or approximated approximately by stationary processes.