Stationary stochastic process

stochastic process, homogeneous in time

2020 Mathematics Subject Classification: Primary: 60G10 [MSN][ZBL]

A stochastic process $ X( t) $ whose statistical characteristics do not change in the course of time $ t $, i.e. are invariant relative to translations in time: $ t \rightarrow t + a $, $ X( t) \rightarrow X( t+ a) $, for any fixed value of $ a $( either a real number or an integer, depending on whether one is dealing with a stochastic process in continuous or in discrete time). The concept of a stationary stochastic process is widely used in applications of probability theory in various areas of natural science and technology, since these processes accurately describe many real phenomena accompanied by unordered fluctuations. For example, the pulsations of the force of a current or the voltage in an electrical chain (electrical "noise" ) can be considered as stationary stochastic processes if the chain is in a stationary system; the pulsations of velocity or pressure at a point of a turbulent flow are stationary stochastic processes if the flow is stationary, etc.

In the mathematical theory of stationary stochastic processes, an important role is played by the moments of the probability distribution of the process $ X( t) $, and especially by the moments of the first two orders — the mean value $ {\mathsf E} X( t) = m $, and its covariance function $ {\mathsf E} [( X( t + \tau ) - {\mathsf E} X( t + \tau )) ( X( t) - EX( t)) ] $, or, equivalently, the correlation function $ E X( t+ \tau ) X( t) = B( \tau ) $. In much of the research into the theory of stationary stochastic processes, the properties that are completely defined by the characteristics $ m $ and $ B( \tau ) $ alone are studied (the so-called correlation theory or theory of second-order stationary stochastic processes). Accordingly, the stochastic processes $ X( t) $ for which $ {\mathsf E} X( t) $ and $ {\mathsf E} X( t + \tau ) X( t) $ do not depend on $ t $ are often separated into a special class and are called stationary stochastic processes in the wide sense. The more special stochastic processes, none of whose characteristics change with time (so that the distribution function $ F _ {t _ {1} \dots t _ {n} } ( x _ {1} \dots x _ {n} ) $ of an $ n $- dimensional random variable $ \{ X( t _ {1} ) \dots X( t _ {n} ) \} $ depends here, for any $ n $, only on the $ n- 1 $ differences $ t _ {2} - t _ {1} \dots t _ {n} - t _ {1} $) are called stationary stochastic processes in the strict sense. Accordingly, the theory of stationary stochastic processes is divided into the theory of stationary stochastic processes in the strict sense and in the wide sense, with a different mathematical apparatus used in each.

In the strict sense, the theory can be stated outside the framework of probability theory as the theory of one-parameter groups of transformations of a measure space that preserve the measure; this theory is very close to the general theory of dynamical systems (cf. Dynamical system) and to ergodic theory. The most important general theorem of the theory of stationary stochastic processes in the strict sense is the Birkhoff–Khinchin ergodic theorem, according to which for any stationary stochastic process $ X( t) $ in the strict sense having a mathematical expectation (i.e. $ {\mathsf E} | X( t) | < \infty $), the limit

$$ \tag{1 } \lim\limits _ {T- S \rightarrow \infty } \frac{1}{T-S} \int\limits _ { S } ^ { T } X( t) dt = \widehat{X} , $$

or

$$ \tag{1a } \lim\limits _ {T- S \rightarrow \infty } \frac{1}{T-S} \sum_{t=S+1}^ { T } X( t) = \widehat{X} , $$

exists with probability 1 (formula (1) relates to processes in continuous time, while (1a) relates to processes in discrete time). There is a result of E.E. Slutskii [Sl], related to stationary stochastic processes in the wide sense, which states that the limit (1) or (1a) exists in mean square. This limit coincides with $ {\mathsf E} X( t) $ if and only if

$$ \tag{2 } \lim\limits _ {T \rightarrow \infty } T ^ {-1} \int\limits _ { 0 } ^ { T } b( \tau ) d \tau = 0 , $$

or

$$ \tag{2a } \lim\limits _ {T \rightarrow \infty } \ T ^ {-1} \sum _ {\tau = 0 } ^ { T-1 } b( \tau ) = 0, $$

where

$$ b( \tau ) = B( \tau ) - m ^ {2} = \ {\mathsf E} [ X( t+ \tau ) - m][ X( t) - m] $$

(the von Neumann ( $ L _ {2} $-) ergodic theorem). These conditions are satisfied, in particular, when $ b( \tau ) \rightarrow 0 $ as $ \tau \rightarrow \infty $. The Birkhoff–Khinchin theorem can be applied to various stationary stochastic processes in the strict sense of the form

$$ Y _ \Phi ( s) = \Phi [ X( t+ s)], $$

where $ \Phi [ X( t)] $ is an arbitrary functional of the stationary stochastic process $ X( t) $, and is a random variable which has a mathematical expectation; if for all such stationary stochastic processes $ Y _ \Phi ( s) $ the corresponding limit $ \widehat{Y} _ \Phi $ coincides with $ {\mathsf E} Y _ \Phi $, then $ X( t) $ is called a metrically transitive stationary stochastic process. For stationary Gaussian stochastic processes $ X( t) $, the condition of being stationary in the strict sense coincides with the condition of being stationary in the wide sense; metric transitivity will occur if and only if the spectral function $ F( \lambda ) $ of $ X( t) $ is a continuous function of $ \lambda $( see, for example, [R], [CL]). There are, in general, no simple necessary and sufficient conditions for the metric transitivity of a stationary stochastic process $ X( t) $.

Apart from the above result relating to metric transitivity, there are also other results specifically for stationary Gaussian stochastic processes. For these processes, detailed studies have been made of the question of the local properties of the realizations of $ X( t) $( i.e. of individual observed values), and of the statistical properties of the sequence of zeros or maxima of the realizations of $ X( t) $, and the points of intersection of it with a given level (see, for example, [CL]). A typical example of the results related to intersections with a level is the statement that, given broad regularity conditions, the set of points of intersection of a high level $ x= u $ with the stationary Gaussian stochastic process $ X( t) $ in a certain special time scale (dependent on $ u $ and tending rapidly to infinity when $ u \rightarrow \infty $) converges to a Poisson flow of events of unit intensity when $ u \rightarrow \infty $( see [CL]).

When studying stationary stochastic processes in the wide sense, the Hilbert space $ H _ {X} $ of linear combinations of values of the process $ X( t) $ and the mean-square limits of sequences of such linear combinations are examined, and a scalar product is defined in it by the formula $ ( Y _ {1} , Y _ {2} ) = {\mathsf E} Y _ {1} Y _ {2} $. In this case, the transformation $ X( t) \rightarrow X( t+ a) $, where $ a $ is a fixed number, will generate a linear unitary operator $ U _ {a} $ mapping the space $ H _ {X} $ onto itself; the family of operators $ U _ {a} $ clearly satisfies the condition $ U _ {a} U _ {b} = U _ {a+} b $, while the values $ X( t) = U _ {t} X( 0) $ form a set of points (a curve if the time $ t $ is continuous, and a countable sequence of points if the time is discrete), mapped onto itself by all operators $ U _ {a} $. Accordingly, the theory of stationary stochastic processes in the wide sense can be reformulated in terms of functional analysis as the study of the sets of points $ X( t) = U _ {t} X( 0) $ of the Hilbert space $ H _ {X} $, where $ U _ {t} $ is a family of linear unitary operators such that $ U _ {a} U _ {b} = U _ {a+} b $( cf. also Semi-group of operators).

Spectral considerations, based on the expansion of the stochastic process $ X( t) $ and its correlation function $ B( \tau ) $ into a Fourier–Stieltjes integral, are central to the theory of stationary stochastic processes in the wide sense. By Khinchin's theorem [Kh] (which is a simple consequence of Bochner's analytic theorem on the general form of a positive-definite function), the correlation function $ B( \tau ) $ of a stationary stochastic process in continuous time can always be represented in the form

$$ \tag{3 } B( \tau ) = \int\limits _ \Lambda e ^ {i \tau \lambda } dF( \lambda ), $$

where $ F( \lambda ) $ is a bounded monotone non-decreasing function of $ \lambda $, while $ \Lambda = (- \infty , \infty ) $; the Herglotz theorem on the general form of positive-definite sequences similarly shows that the same representation, but with $ \Lambda = [- \pi , \pi ] $, also holds for the correlation function of a stationary stochastic process in discrete time. If the correlation function $ B( \tau ) $ decreases sufficiently rapidly as $ | \tau | \rightarrow \infty $( as is most often the case in applications under the condition that by $ X( t) $ one understands the difference $ X( t)- m $, i.e. it is considered that $ {\mathsf E} X( t) = 0 $), then the integral at the right-hand side of (3) becomes an ordinary Fourier integral

$$ \tag{4 } B( \tau ) = \int\limits _ \Lambda e ^ {i \tau \lambda } f( \lambda ) d \lambda , $$

where $ f( \lambda ) = F ^ { \prime } ( \lambda ) $ is a non-negative function. The function $ F( \lambda ) $ is called the spectral function of $ X( t) $, while the function $ f( \lambda ) $( in cases where the equality (4) holds) is called its spectral density. Starting from the Khinchin formula (3) (or from the definition of the process $ X( t) $ in the form of the set of points $ X( t) = U _ {t} X( 0) $ in the Hilbert space $ H _ {X} $, and Stone's theorem on the spectral representation of one-parameter groups of unitary operators in a Hilbert space), it can also be demonstrated that the process $ X( t) $ itself permits a spectral decomposition in the form

$$ \tag{5 } X( t) = \int\limits _ \Lambda e ^ {it \lambda } dZ( \lambda ), $$

where $ Z( \lambda ) $ is a random function with uncorrelated increments (i.e. $ {\mathsf E} dZ( \lambda _ {1} ) dZ( \lambda _ {2} ) = 0 $ when $ \lambda _ {1} \neq \lambda _ {2} $) which satisfies the condition $ {\mathsf E} | dZ( \lambda ) | ^ {2} = dF( \lambda ) $, while the integral at the right-hand side is understood to be the mean-square limit of the corresponding sequence of integral sums. The decomposition (5) provides grounds for considering any stationary stochastic process in the wide sense as a superposition of a set of non-correlated harmonic oscillations of different frequencies with random amplitudes and phases; the spectral function $ F( \lambda ) $ and the spectral density $ f( \lambda ) $ define the distribution of the average energy (or, more accurately, of the power) of the harmonic oscillations with frequency spectrum $ \lambda $ that constitute $ X( t) $( as a result of which the function $ f( \lambda ) $ in applied research is often called the energy spectrum, or power spectrum, of $ X( t) $).

The spectral decomposition of the correlation function $ B( \tau ) $, defined by formula (3), demonstrates that the mapping $ X( t) \rightarrow e ^ {it \lambda } $, which maps elements $ X( t) $ of the Hilbert space $ H _ {X} $ to elements $ e ^ {it \lambda } $ of the Hilbert space $ L _ {2} ( dF ) $ of complex-valued functions on the set $ \Lambda $ with a modulus, square-integrable with respect to $ dF( \lambda ) $, is an isometric mapping of $ H $ into $ L _ {2} ( dF ) $. This mapping can be extended to an isometric linear mapping $ M $ of the whole space $ H _ {X} $ onto the space $ L _ {2} ( dF ) $, a fact that allows one to reformulate many problems in the theory of stationary stochastic processes in the wide sense as problems in function theory.

A significant part of the theory of stationary stochastic processes in the wide sense is devoted to methods of solving linear approximation problems for such processes, i.e. methods of locating a linear combination of any "known" values of $ X( t) $ that best approximates (in the sense of the minimum least-square error) a certain "unknown" value of the same process or any "unknown" random variable $ Y $. In particular, the problem of optimal linear extrapolation of $ X( t) $ consists of finding the best approximation $ X ^ {*} ( s) $ of the value $ X( s) $, $ s > 0 $, that linearly depends on the "past values" of $ X( t) $ with $ t \leq 0 $; the problem of optimal linear interpolation consists of finding the best approximation for $ X( s) $ that linearly depends on the values of $ X( t) $, where $ t $ runs through all values that do not belong to a specific interval of the time axis (to which $ s $ does belong); the problem of optimal linear filtering can be formulated as the problem of finding the best approximation $ Y ^ {*} $ for a certain random variable $ Y $( which is usually the value for some $ t= s $ of a stationary stochastic process $ Y( t) $, correlated with $ X( t) $, whereby $ Y( t) $ most often plays the part of a "signal" , while $ X( t) = Y( t) + N( t) $ is the sum of the "signal" and a "noise" $ N( t) $ that interferes with it, and the sum is known from the observations) that linearly depends on the values of $ X( t) $ when $ t \leq 0 $( see Stochastic processes, prediction of; Stochastic processes, filtering of; Stochastic processes, interpolation of).

All these problems reduce geometrically to the problem of projecting a point of the Hilbert space $ H _ {X} $( or of its extension) orthogonally onto a given subspace of this space. Relying on this geometric interpretation and on the isomorphism of the spaces $ H _ {X} $ and $ L _ {2} ( dF ) $, A.N. Kolmogorov has deduced general formulas that make it possible to determine the mean-square error of optimal linear extrapolation or interpolation, corresponding to the case where the value of $ X( t) $ is unknown only when $ t= s $, by means of the spectral function $ F( \lambda ) $ of the stationary stochastic process $ X( t) $ in discrete time $ t $( see [R], [Ko]–[D]). When used for the extrapolation problem, the same results were obtained for processes $ X( t) $ in continuous time by M.G. Krein and K. Karhunen. N. Wiener [W] demonstrated that the search for the best approximation $ X ^ {*} ( s) $ or $ Y ^ {*} = Y ^ {*} ( s) $ in the case of problems of optimal linear extrapolation and filtering can be reduced to the solution of a certain integral equation of Wiener–Hopf type, or (when $ t $ is discrete) of the discrete analogue of such an equation, which makes it possible to use the factorization method (see Wiener–Hopf equation; Wiener–Hopf method). Problems of optimal linear extrapolation or filtering of a stationary stochastic process $ X( t) $ in continuous time in the case where not all its past values for $ t \leq 0 $ are known but only its values on a finite interval $ - T \leq t \leq 0 $, as well as the problem of optimal linear interpolation of such an $ X( t) $, can be reduced to certain problems of establishing a special form of differential equation (a "generalized string equation" ) by means of its spectrum (see [Kr], [DM]).

The above approaches to the solution of problems of optimal linear extrapolation, interpolation and filtering provide sufficiently-simple explicit formulas for the required best approximation $ X ^ {*} ( s) $ or $ Y ^ {*} $ that can be successfully used in practice only in certain exceptional cases. One important case in which such explicit formulas do exist is the case of a stationary stochastic process $ X( t) $ with rational spectral density $ f( \lambda ) $ relative to the $ e ^ {i \lambda } $( if $ t $ is discrete) or relative to $ \lambda $( if $ t $ is continuous), which was studied in detail by Wiener [W] (for applications to problems of extrapolation and filtering by values where $ t \leq 0 $). It was subsequently demonstrated that for such stationary stochastic processes with rational spectral density there is also an explicit solution of the problems of linear interpolation, extrapolation and filtering by means of data on a finite interval $ - T \leq t \leq 0 $( see, for example, [R], [Y]). The simplicity of processes with rational spectral density can be explained by the fact that such stationary stochastic processes (and practically only they) are a one-dimensional component of a multi-dimensional stationary Markov process (see [D2]).

The concept of a stationary stochastic process permits a whole series of generalizations. One of these is the concept of a generalized stationary stochastic process. This is a generalized stochastic process (cf. Stochastic process, generalized) $ X( \phi ) $( i.e. a random linear functional, defined on the space $ D $ of infinitely-differentiable functions $ \phi ( t) $ with compact support), such that either the distribution function of the random vector $ \{ X( V _ {a} \phi _ {1} ) \dots X( V _ {a} \phi _ {n} ) \} $, where $ V _ {a} \phi ( t) = \phi ( t+ a) $ for any positive integer $ n $, real number $ a $ and $ \phi _ {1} \dots \phi _ {2} \in D $, coincides with the probability distribution of the vector $ \{ X( \phi _ {1} ) \dots X( \phi _ {n} ) \} $( a generalized stationary stochastic process in the strict sense), or else

$$ {\mathsf E} X( \phi ) = {\mathsf E} X( V _ {a} \phi ), $$

$$ {\mathsf E} X( V _ {a} \phi _ {1} ) \overline{ {X( V _ {a} \phi _ {2} ) }}\; = {\mathsf E} X( \phi _ {1} ) \overline{ {X( \phi _ {2} ) }}\; , $$

for all $ a $( a generalized stationary stochastic process in the wide sense). A generalized stationary stochastic process in the wide sense $ X( \phi ) $ and its correlation functional $ B( \phi _ {1} , \phi _ {2} ) = {\mathsf E} X( \phi _ {1} ) \overline{ {X( \phi _ {2} ) }}\; $( or covariance functional $ {\mathsf E} [ ( X( \phi _ {1} ) - {\mathsf E} X( \phi _ {1} )) \overline{ {( X( \phi _ {2} ) - {\mathsf E} X( \phi _ {2} )) }}\; ] $) permit a spectral decomposition related to (2) and (5) (see Spectral decomposition of a random function). Other frequently-used generalizations of the concept of a stationary stochastic process are the concepts of a stochastic process with stationary increments of a certain order and of a homogeneous random field (cf. Random field, homogeneous).

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In the English language literature one says ergodic stationary process rather than metrically-transitive stationary process. See also Gaussian process.

The terminology concerning the phrases "correlation function" and "covariance function" is not yet completely standardized. In probability theory the following terminology seems just about universally adopted. Given two random variables $ X, Y $ their covariance is $ \mathop{\rm cov} ( X, Y) = {\mathsf E} [( X- {\mathsf E} X )( Y- {\mathsf E} Y )] $, their correlation coefficient is $ \rho ( X, Y) = \mathop{\rm cov} ( X, Y) / ( \sigma ( X) \sigma ( Y)) $, where $ \sigma ( X) $ is the variance of $ X $, $ \mathop{\rm var} ( X) = \mathop{\rm cov} ( X, X) $( cf. Dispersion), and there is no special name for the mixed second-order moment $ E( XY) $. Correspondly one has the terms covariance function and correlation function, also termed auto-correlation function and auto-covariance function, for the following quantities associated to a stochastic process:

$$ B( s, t) = {\mathsf E} [( X( s) - {\mathsf E} X( s)) ( X( t)- {\mathsf E} X( t)) ] $$

$$ \rho ( s, t) = \sigma ( X( s)) ^ {-1} \sigma ( X( t)) ^ {-1} B( s, t) , $$

and for two stochastic processes $ X( t), Y( t) $ there are correspondingly the cross covariance function and cross correlation function

$$ B _ {XY} ( s, t ) = \ {\mathsf E} [( X( s) - {\mathsf E} X( s)) ( Y( t) - {\mathsf E} Y( t)) ] , $$

$$ \rho _ {XY} ( s, t) = \sigma ( X( s)) ^ {-1} \sigma ( Y( t)) ^ {-1} B _ {XY} ( s, t). $$

However, in application areas of probability theory a somewhat different terminology is also employed, for instance in oceanology, hydrology and electrical engineering (cf. [J]–[So]). E.g., one also finds the phrase "correlation function" for the quantity $ B( s, t) $( instead of covariance function). Also, one regularly finds the terminology "correlation function" for the quantity

$$ \gamma ( s, t) = {\mathsf E} [ X( s) X( t)] , $$

which is also in agreement with the phrase "correlation function" (pair correlation function) as it is used in statistical mechanics, cf. e.g. [T].

If the process is stationary, the distinctions are minor. Thus, if $ m = {\mathsf E} X( s) $, $ \sigma = \sigma ( X( s)) $, $ B( s, t) = \gamma ( s, t) - m ^ {2} $, $ \rho ( s, t) = \sigma ^ {-2} B( s, t) $.

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