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A function of an arbitrary argument  $  t $(
defined on the set  $  T $
of its values, and taking numerical values or, more generally, values in a vector space) whose values are defined in terms of a certain experiment and may vary with the outcome of this experiment according to a given probability distribution. In [[Probability theory|probability theory]], attention centres on numerical (that is, scalar) random functions  $  X ( t) $; 
a random vector function  $  \mathbf X ( t) $
can be regarded as the aggregate of the scalar functions  $  X _  \alpha  ( t) $, 
where  $  \alpha $
ranges over the finite or countable set  $  A $
of components of  $  \mathbf X $, 
that is, as a numerical random function on the set  $  T _ {1} = T \times A $
of pairs  $  ( t , \alpha ) $, 
$  t \in T $, 
$  \alpha \in A $.

When  $  T $
is finite,  $  X ( t) $
is a finite set of random variables, and can be regarded as a multi-dimensional (vector) random variable characterized by a multi-dimensional distribution function. When  $  T $
is infinite, the case mostly studied is that in which  $  t $
takes numerical (real) values; in this case,  $  t $
usually denotes time, and  $  X ( t) $
is called a [[Stochastic process|stochastic process]], or, if  $  t $
takes only integral values, a [[Random sequence|random sequence]] (or time series). If the values of  $  t $
are the points of a manifold (such as a  $  k $-
dimensional Euclidean space  $  \mathbf R  ^ {k} $), 
then  $  X ( t) $
is called a [[Random field|random field]].

The probability distribution of the values of a random function  $  X ( t) $
defined on an infinite set  $  T $
is characterized by the aggregate of finite-dimensional probability distributions of sets of random variables  $  X ( t _ {1} ) \dots X ( t _ {n} ) $
corresponding to all finite subsets  $  \{ t _ {1} \dots t _ {n} \} $
of  $  T $, 
that is, the aggregate of corresponding finite-dimensional distribution functions  $  F _ {t _ {1}  \dots t _ {n} } ( x _ {1} \dots x _ {n} ) $, 
satisfying the consistency conditions:

$$ \tag{1 }
F _ {t _ {1}  \dots t _ {n} , t _ {n+} 1 \dots t _ {n+} m } ( x _ {1} \dots x _ {n} , \infty \dots \infty ) =
$$

$$ 
= \ 
F _ {t _ {1}  \dots t _ {n} } ( x _ {1} \dots x _ {n} ) ,
$$

$$ \tag{2 }
F _ {t _ {i _ {1}  } \dots t _ {i _ {n}  } }
( x _ {i _ {1}  } \dots x _ {i _ {n}  } )  =  F _ {t _ {1}  \dots t _ {n} } ( x _ {1} \dots x _ {n} ) ,
$$

where  $  i _ {1} \dots i _ {n} $
is an arbitrary permutation of the subscripts  $  1 \dots n $. 
This characterization of the probability distribution of  $  X ( t) $
is sufficient in all cases when one is only interested in events depending on the values of  $  X $
on countable subsets of  $  T $. 
But it does not enable one to determine the probability of properties of  $  X $
that depend on its values on a continuous subset of  $  T $, 
such as the probability of continuity or differentiability, or the probability that  $  X ( t) < a $
on a continuous subset of  $  T $(
see [[Separable process|Separable process]]).

Random functions can be described more generally in terms of aggregates of random variables  $  X = X ( \omega ) $
defined on a fixed [[Probability space|probability space]]  $  ( \Omega , {\mathcal A} , {\mathsf P} ) $(
where  $  \Omega $
is a set of points  $  \omega $, 
$  {\mathcal A} $
is a  $  \sigma $-
algebra of subsets of  $  \Omega $
and  $  {\mathsf P} $
is a given probability measure on  $  {\mathcal A} $), 
one for each point  $  t $
of  $  T $. 
In this approach, a random function on  $  T $
is regarded as a function  $  X ( t , \omega ) $
of two variables  $  t \in T $
and  $  \omega \in \Omega $
which is  $  {\mathcal A} $-
measurable for every  $  t $(
that is, for fixed  $  t $
it reduces to a random variable defined on the probability space  $  ( \Omega , {\mathcal A} , {\mathsf P} ) $). 
By taking a fixed value  $  \omega _ {0} $
of  $  \omega $, 
one obtains a numerical function  $  X ( t , \omega _ {0} ) = x ( t) $
on  $  T $, 
called a realization (or sample function or, when  $  t $
denotes time, a trajectory) of  $  X ( t) $; 
$  {\mathcal A} $
and  $  {\mathsf P} $
induce a  $  \sigma $-
algebra of subsets and a probability measure defined on it in the function space  $  \mathbf R  ^ {T} = \{ {x ( t) } : {t \in T } \} $
of realizations  $  x ( t) $, 
whose specification can also be regarded as equivalent to that of the random function. The specification of a random function as a probability measure on a  $  \sigma $-
algebra of subsets of the function space  $  \mathbf R  ^ {T} $
of all possible realizations  $  x ( t) $
can be regarded as a special case of its general specification as a function of two variables  $  X ( t , \omega ) $(
where  $  \omega $
belongs to the probability space  $  ( \Omega , {\mathcal A} , {\mathsf P} ) $
in which  $  \Omega = \mathbf R  ^ {T} $), 
that is, elementary events (points  $  \omega $
in the given probability space) are identified at the outset with the realizations  $  x ( t) $
of  $  X ( t) $. 
On the other hand, it is also possible to show that any other way of specifying  $  X ( t) $
can be reduced to this form using a special determination of a probability measure on  $  \mathbf R  ^ {T} $. 
In particular, Kolmogorov's fundamental theorem on consistent distributions (see [[Probability space|Probability space]]) shows that the specification of the aggregate of all possible finite-dimensional distribution functions  $  F _ {t _ {1}  \dots t _ {n} } ( x _ {1} \dots x _ {n} ) $
satisfying the above consistency conditions (1) and (2) defines a probability measure on the  $  \sigma $-
algebra of subsets of the function space  $  \mathbf R  ^ {T} = \{ {x ( t) } : {t \in T } \} $
generated by the aggregate of cylindrical sets (cf. [[Cylinder set|Cylinder set]]) of the form  $  \{ {x ( t) } : {[ x ( t _ {1} ) \dots x ( t _ {n} ) ] \in B  ^ {n} } \} $, 
where  $  n $
is an arbitrary positive integer and  $  B  ^ {n} $
is an arbitrary Borel set of the  $  n $-
dimensional space  $  \mathbf R  ^ {n} $
of vectors  $  [ x ( t _ {1} ) \dots x ( t _ {n} ) ] $.

For references see [[Stochastic process|Stochastic process]].

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Doob,   "Stochastic processes" , Wiley  (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Loève,   "Probability theory" , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "The theory of stochastic processes" , '''1''' , Springer  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Blanc-Lapierre,   R. Fortet,   "Theory of random functions" , '''1–2''' , Gordon &amp; Breach  (1965)  (Translated from French)</TD></TR></table>