A process (that is, a variation with time of the state of
a certain system) whose course depends on chance and for which probabilities
for some courses are given. A typical example of this is
Brownian motion.
Other examples of practical importance are: the fluctuation of current in
an electrical circuit in the presence of so-called thermal noise,
the random changes in the level of received radio-signals
in the presence of random weakening of radio-signals (fading)
created by meteorological or other disturbances, and the turbulent flow
of a liquid or gas. To these can be added many industrial
processes accompanied by random fluctuations, and also certain processes encountered
in geophysics (e.g., variations of the Earth's magnetic
field, unordered sea-waves and microseisms, that is, high-frequency
irregular oscillations of the level of the surface
of the Earth), biophysics (for example, variations of the
bio-electric potential of the brain registered
on an electro-encephalograph), and economics.
The mathematical theory of stochastic processes regards the instantaneous state
of the system in question as a point of a certain phase space
(the space of states), so that the stochastic process is a function
of the time
with values in
.
It is usually assumed that
is a vector space, the most studied case (and the most
important one for applications) being the narrower one where the points of
are given by one or more numerical parameters (a
generalized coordinate system). In the narrow case a stochastic
process can be regarded either simply as a numerical function
of time taking various values depending on chance (i.e. admitting various realizations
,
a
one-dimensional stochastic process),
or similarly as a vector function
(a
multi-dimensional
or
vector stochastic process).
The study of multi-dimensional stochastic processes can
be reduced to that of one-dimensional stochastic processes by passing from
to an auxiliary process
where
is an arbitrary
-dimensional
vector. Therefore the study of one-dimensional processes occupies a central
place in the theory of stochastic processes. The parameter
usually takes arbitrary real values or values in an interval on the real axis
(when one wishes to stress this, one speaks of a
stochastic process in continuous time),
but it may take only integral values, in which case
is called a
stochastic process in discrete time
(or a
random sequence
or a
time series).
The representation of a probability distribution in the infinite-dimensional
space of all variants of the course of
(that is, in the space of realizations
)
does not fall within the scope of the classical methods
of probability theory and requires the construction of
a special mathematical apparatus. The only exceptions
are special classes of stochastic processes whose
probabilistic nature is completely determined by the dependence of
on a certain finite-dimensional random vector
,
since in this case the probability of the course followed by
depends only on the finite-dimensional probability distribution of
.
An example of a stochastic process of this type which is
of practical importance is a random harmonic oscillation of the form
where
is a fixed number and
and
are independent random variables. This process is often used
in the investigation of amplitude-phase modulation in radio-technology.
A wide class of probability distributions for stochastic
processes is characterized by an infinite family
of compatible finite-dimensional probability distributions of the random vectors
corresponding to all finite subsets
of values of
(see
Random function).
However, knowledge of all these distributions is not
sufficient to determine the probabilities of events depending on the values of
for an uncountable set of values of
,
that is, it does not determine the stochastic process
uniquely.
Example.
Let
,
,
be a harmonic oscillation with random phase
.
Let a random variable
be uniformly distributed on the interval
,
and let
,
,
be the stochastic process given by the equations
when
,
when
.
Since
for any fixed finite set of points
,
it follows that all the finite-dimensional distributions of
and
are identical. At the same time,
and
are different: in particular, all realizations of
are continuous (having sinusoidal form), while all realizations of
have a point of discontinuity, and all realizations of
do not exceed 1, but no realization of
has this property. Hence it follows that a
given system of finite-dimensional probability distributions can correspond to
distinct modifications of a stochastic process, and one
cannot compute, purely from knowledge of this system, either
the probability that a realization of the stochastic process
will be continuous, or the probability that it will be bounded by some fixed constant.
However, from knowledge of all finite-dimensional probability
distributions one can often clarify whether or not there exists a stochastic process
that has these finite-dimensional distributions, and is such
that its realizations are continuous (or differentiable
or nowhere exceed a given constant
)
with probability 1. A typical example of a general
condition guaranteeing the existence of a stochastic process
with continuous realizations with probability 1
and given finite-dimensional distributions is
Kolmogorov's condition:
If the finite-dimensional probability distributions of a stochastic process
,
defined on the interval
,
are such that for some
,
,
,
and all sufficiently small
,
the following inequality holds:
(which evidently imposes restrictions only on the two-dimensional distributions of
),
then
has a modification with continuous realizations with probability 1 (see
[1]–
[6],
for example). In the special case of a Gaussian process
,
condition
(1)
can be replaced by the weaker condition
for some
,
,
.
This holds with
and
for the
Wiener process
and the
Ornstein–Uhlenbeck process,
for example. In cases where, for given finite-dimensional
probability distributions, there is a modification of
whose realizations are continuous (or differentiable or bounded by a constant
)
with probability 1, all other modifications of the same process
can usually be excluded from consideration by requiring that
satisfies a certain very general regularity condition,
which holds in almost-all applications (see
Separable process).
Instead of specifying the infinite system of
finite-dimensional probability distributions of a stochastic process
,
this can be defined using the values of the corresponding
characteristic functional
where
ranges over a sufficiently wide class of linear functionals depending on
.
If
is continuous in probability for
(that is,
as
for any
)
and
is a function of bounded variation on
,
then
is a random variable. One may take
in
(3),
where
is denoted by the symbol
for convenience. In many cases it is sufficient to consider only linear functionals
of the form
where
is an infinitely-differentiable function of compact support in
(and the interval
may be taken finite). Under fairly general regularity conditions, the values
uniquely determine all finite-dimensional probability distributions of
,
since
where
is the characteristic function of the random vector
,
as
(here
is the Dirac
-function,
and convergence is understood in the sense of convergence of generalized functions). If
does not tend to a finite limit, then
has no finite values at any fixed point and only smoothed values
have a meaning, that is, the characteristic functional
does not give an ordinary
( "classical" )
stochastic process
,
but a generalized stochastic process (cf.
Stochastic process, generalized)
.
The problem of describing all finite-dimensional probability distributions of
is simplified in those cases when they are all uniquely determined
by the distributions of only a few lower orders. The
most important class of stochastic processes for which
all multi-dimensional distributions are determined by the
values of the one-dimensional distributions of
are sequences of independent random variables (which are
special stochastic processes in discrete time). Such processes
can be studied within the framework of classical probability
theory, and it is important that some important
classes of stochastic processes can be effectively specified as functions of a sequence
,
of independent random variables. For example, the
following stochastic processes are of significant interest:
or
(see
Moving-average process),
and
where
,
is a prescribed system of functions on the interval
(see
Spectral decomposition of a random function).
Three important classes of stochastic processes are described
below, for which all finite-dimensional distributions are
determined by the one-dimensional distributions of
and the two-dimensional distributions of
.
1)
The class of stochastic processes with independent increments (cf.
Stochastic process with independent increments)
,
for which
and
are independent variables
(
).
To represent
on the interval
it is convenient to use the distribution functions
and
,
where
,
of the random variables
and
,
in which case
must evidently satisfy the functional equation
Using
(4)
it is possible to show that if
is continuous in probability, then its characteristic functional
can be written in the form
where
is a continuous function,
is a non-decreasing continuous function such that
and
is an increasing continuous measure on
in
.
2)
The class of Markov processes
for which, when
,
the conditional probability distribution of
given all values of
for
depends only on
.
To represent a
Markov process
,
,
it is convenient to use the distribution function
of the value
and the transition function
,
which is defined for
as the conditional probability that
given that
.
The function
must satisfy the
Kolmogorov–Chapman equation,
similar to
(4),
and this enables one, under certain
conditions, to obtain the simpler forward and backward
Kolmogorov equation
(e.g. the Fokker–Planck equation) for this function.
3)
The class of Gaussian processes
for which all multi-dimensional probability distributions of the vectors
are Gaussian (normal) distributions. Since a normal distribution is
uniquely determined by its first and second moments, a
Gaussian process
is determined by the values of the functions
and
where
must be a non-negative definite kernel such that
is a non-negative definite kernel. The characteristic functional
of a Gaussian process
,
where
,
is
4)
Another important class of stochastic processes
is that of stationary stochastic processes
,
where the statistical characteristics do not change in the course
of time, that is, they are invariant under the transformation
,
for any fixed number
.
The multi-dimensional probability distributions of a general
stationary stochastic process
cannot be described in a simple manner, but for many problems concerning
such processes it is sufficient to know only the values of the first two moments,
and
(so that here the only necessary assumption is
of stationarity in the wide sense, i.e. the moments
and
are independent of
).
It is essential that any stationary stochastic process (at least in
the wide sense) admits a spectral decomposition of the form
where
is a stochastic process with non-correlated increments. In particular, it follows that
where
is the monotone non-decreasing spectral function of
(cf.
Spectral function of a stationary stochastic process).
The spectral decompositions
(5)
and
(6)
lie at the heart of the
solution of problems of best (in the sense
of minimal mean-square error) linear extrapolation, interpolation
and filtering of stationary stochastic processes.
The mathematical theory of stochastic processes also includes a
large number of results related to a series of
subclasses or, conversely, of extensions, of the
above classes of stochastic processes (see
Markov chain;
Diffusion process;
Branching process;
Martingale;
Stochastic process with stationary increments;
etc.).