Random process

A random process (a probabilistic process, a random function, a stochastic process) in probability theory is a family of random variables indexed by some parameter , most often playing the role of time or coordinate .

Content

Definition

Let probability space be given $(\Omega ,{\mathcal {F}},\mathbb {P} )$ ${\ displaystyle (\ Omega, {\ mathcal {F}}, \ mathbb {P})}$ . Parameterized Family $\{X_{t}\}_{t\in T}$ ${\ displaystyle \ {X_ {t} \} _ {t \ in T}}$ random variables

X_{t}(\cdot )\colon \Omega \to \mathbb {R} ,\quad t\in T

{\ displaystyle X_ {t} (\ cdot) \ colon \ Omega \ to \ mathbb {R}, \ quad t \ in T}

,

Where $T$ ${\ displaystyle T}$ an arbitrary set is called a random function .

One-dimensional random process $X(t)$ ${\ displaystyle X (t)}$ flowing in time $t$ ${\ displaystyle t}$ , is called a process whose value for any fixed $t=t_{0}$ ${\ displaystyle t = t_ {0}}$ is a random variable $X(t_{0})$ ${\ displaystyle X (t_ {0})}$ ^[1] .

Terminology

If a $T\subset \mathbb {R}$ ${\ displaystyle T \ subset \ mathbb {R}}$ , then the parameter $t\in T$ ${\ displaystyle t \ in T}$ can be interpreted as time . Then the random function $\{X_{t}\}$ ${\ displaystyle \ {X_ {t} \}}$ called a random process . If the set $T$ ${\ displaystyle T}$ discrete, for example $T\subset \mathbb {N}$ ${\ displaystyle T \ subset \ mathbb {N}}$ , then such a random process is called a random sequence .

If a $T\subset \mathbb {R} ^{n}$ ${\ displaystyle T \ subset \ mathbb {R} ^ {n}}$ where $n\geqslant 1$ ${\ displaystyle n \ geqslant 1}$ , then the parameter $t\in T$ ${\ displaystyle t \ in T}$ can be interpreted as a point in space, and then a 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 $X(t)$ ${\ displaystyle X (t)}$ a process is called discrete in time if the system in which it flows changes its states only at times $\;t_{1},t_{2},\ldots$ ${\ displaystyle \; t_ {1}, t_ {2}, \ ldots}$ 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 time instants $\;t_{1},t_{2},\ldots ,t_{n}$ ${\ displaystyle \; t_ {1}, t_ {2}, \ ldots, t_ {n}}$ , but not from the values themselves of these quantities. In other words, a random process is called stationary if its probabilistic laws are constant over time. Otherwise, it is called non-stationary .
A random function is called stationary in the broad sense if its mathematical expectation and dispersion 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 was introduced by A. Ya. Khinchin .
A random process is called a process with stationary increments of a certain order if the probabilistic laws of such an increment are constant in time. Such processes were considered by Yaglom ^[2] .
If the ordinates of a random function obey the normal distribution law , then the function itself is called normal .
Random functions whose distribution law of ordinates at a future moment in time is completely determined by the value of the ordinate of the process at a given moment of time and does not depend on the values of the ordinates of the process at previous times are called Markovian .
A random process is called a process with independent increments if for any set $t_{1},t_{2},\ldots ,t_{n}$ ${\ displaystyle t_ {1}, t_ {2}, \ ldots, t_ {n}}$ where $n>2$ ${\ displaystyle n> 2}$ , but $t_{1}<t_{2}<\ldots <t_{n}$ ${\ displaystyle t_ {1} <t_ {2} <\ ldots <t_ {n}}$ random variables $(X_{t_{2}}-X_{t_{1}})$ ${\ displaystyle (X_ {t_ {2}} - X_ {t_ {1}})}$ , $(X_{t_{3}}-X_{t_{2}})$ ${\ displaystyle (X_ {t_ {3}} - X_ {t_ {2}})}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $(X_{t_{n}}-X_{t_{n-1}})$ ${\ displaystyle (X_ {t_ {n}} - X_ {t_ {n-1}})}$ independent in aggregate.
If in determining the moment functions of a stationary random process, the averaging operation over a statistical ensemble can be replaced by time averaging, then such a stationary random process is called ergodic .
Among random processes, pulsed random processes are distinguished.

Random Process

Let a random process be given $\{X_{t}\}_{t\in T}$ ${\ displaystyle \ {X_ {t} \} _ {t \ in T}}$ . Then for each fixed $t\in T$ ${\ displaystyle t \ in T}$ $X_{t}$ ${\ displaystyle X_ {t}}$ Is a random variable called a section . If an elementary outcome is fixed $\omega \in \Omega$ ${\ displaystyle \ omega \ in \ Omega}$ then $X_{t}\colon T\to \mathbb {R}$ ${\ displaystyle X_ {t} \ colon T \ to \ mathbb {R}}$ - deterministic function of the parameter $t$ ${\ displaystyle t}$ . Such a function is called a trajectory or implementation of a random function. $\{X_{t}\}$ ${\ displaystyle \ {X_ {t} \}}$ .

Examples

$\{X_{n}\}_{n\in \mathbb {N} }$ ${\ displaystyle \ {X_ {n} \} _ {n \ in \ mathbb {N}}}$ where $\;X_{i}\sim \mathrm {N} (0,1)$ ${\ displaystyle \; X_ {i} \ sim \ mathrm {N} (0,1)}$ called the standard Gaussian (normal) random sequence.
Let be $f\colon \mathbb {R} \to \mathbb {R}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$ , and $Y$ ${\ displaystyle Y}$ - random value. Then

X_{t}(\omega )=f(t)\cdot Y(\omega )

{\ displaystyle X_ {t} (\ omega) = f (t) \ cdot Y (\ omega)}

is a random process.

Notes

↑ Wentzel, 1991 , p. 12.
↑ Yaglom A. M. Correlation theory of processes with random stationary parametric increments // Matematicheskii Sbornik. T. 37. Issue. 1, pp. 141-197. - 1955.

Sources

A. A. Sveshnikov. Applied methods of the theory of random functions. - Gl.red.phys.-math. Lit., 1968.
S. I. Baskakov. Radio / technical circuits and signals. - High School, 2000.
Nathan A.A. , Gorbachev O.G., Guz S.A. Fundamentals of the theory of random processes : textbook. manual on the course "Random Processes" - M .: MZ Press - MIPT, 2003. –168 p. ISBN 5-94073-055-8 .
Ventzel E.S. , Ovcharov L.A. Theory of random processes and its engineering applications. - M .: Nauka, 1991 .-- 384 p. - ISBN 5-02-014125-9 .

[_2daa6f7fb98d7f59-1] Wentzel, 1991 , p. 12.

[2] Yaglom A. M. Correlation theory of processes with random stationary parametric increments // Matematicheskii Sbornik. T. 37. Issue. 1, pp. 141-197. - 1955.