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A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902501.png" /> in discrete or continuous time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902502.png" /> such that the statistical characteristics of its increments of some fixed order do not vary with time (that is, are invariant with respect to the time shifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902503.png" />). As in the case of stationary stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]), one distinguishes two types of such processes, namely stochastic processes with stationary increments in the strict sense, for which all finite-dimensional probability distributions of increments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902504.png" /> of a given order at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902506.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902507.png" /> coincide with one another, and stochastic processes with stationary increments in the wide sense, for which the mean values of an increment at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902508.png" /> and the second moments of the increments at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s0902509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025010.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025011.png" />.
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In the case of processes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025012.png" /> in discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025013.png" /> one can always pass from the consideration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025014.png" /> to that of the new stochastic process
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025015.png" /></td> </tr></table>
+
{{MSC|60G99|60G10}}
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025016.png" /> are binomial coefficients. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025017.png" /> is a stochastic process with stationary increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025018.png" />, then the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025019.png" /> is stationary in the usual sense. Thus, in the case of discrete time, the theory of stochastic processes with stationary increments reduces easily to that of the more particular stationary stochastic processes. However, from the point of view of applications, the use of the concept of a stochastic process with stationary increments and discrete time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025020.png" /> often turns out to be very convenient, since for many explicit non-stationary time series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025022.png" /> met in practice, the series of their increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025023.png" /> of some order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025024.png" /> can be regarded as realizations of a stationary stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025025.png" />. In particular, G. Box and G. Jenkins showed in [[#References|[1]]] that, when solving many practical problems, real time series can often be regarded as realizations of a so-called [[Auto-regressive process|auto-regressive process]], an integrated moving-average process that represents a special stochastic process with stationary increments and discrete time (see also [[#References|[2]]]–[[#References|[4]]]).
+
[[Category:Stochastic processes]]
  
Examples of stochastic processes with stationary increments of the first order (in the strict sense) and in continuous time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025026.png" /> are a [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]]. Both of these also belong to the narrower class of processes with independent increments of the first order. In the case of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025027.png" />, the theory of stochastic processes with stationary increments does not reduce directly to the theory of the simpler stationary processes. The correlation theory of stochastic processes with stationary increments of the first order (that is, the theory of the corresponding processes in the wide sense), was developed by A.N. Kolmogorov [[#References|[5]]] (see also [[#References|[6]]]). An analogous theory of stochastic processes with stationary increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025029.png" /> is an arbitrary positive integer, was considered in [[#References|[7]]]–[[#References|[9]]]. A central position in the correlation theory of stochastic processes with stationary increments is occupied by the derivation of the spectral decomposition of such processes and of their second-order moments. The concept of a generalized stochastic process (cf. [[Stochastic process, generalized|Stochastic process, generalized]]) can be used to simplify the theory of stochastic processes with stationary increments. Since in the theory of generalized stochastic processes, any stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025030.png" /> has derivatives of all orders (which are again generalized stochastic processes), a stochastic process with stationary increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025031.png" /> can be defined as a stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025032.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025033.png" />-th derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025034.png" /> is an (in general, generalized) stationary stochastic process (see [[#References|[9]]]).
+
A [[Stochastic process|stochastic process]]  $  X ( t) $
 +
in discrete or continuous time  $  t $
 +
such that the statistical characteristics of its increments of some fixed order do not vary with time (that is, are invariant with respect to the time shifts  $  t \mapsto t + a $).
 +
As in the case of stationary stochastic processes (cf. [[Stationary stochastic process|Stationary stochastic process]]), one distinguishes two types of such processes, namely stochastic processes with stationary increments in the strict sense, for which all finite-dimensional probability distributions of increments of  $  X ( t) $
 +
of a given order at the points  $  t _ {1} \dots t _ {n} $
 +
and  $  t _ {1} + a \dots t _ {n} + a $
 +
for any  $  a $
 +
coincide with one another, and stochastic processes with stationary increments in the wide sense, for which the mean values of an increment at  $  t $
 +
and the second moments of the increments at  $  t $
 +
and  $  t+ s $
 +
do not depend on  $  t $.
 +
 
 +
In the case of processes  $  X ( t) $
 +
in discrete time  $  t = 0 , \pm  1 \dots $
 +
one can always pass from the consideration of  $  X ( t) $
 +
to that of the new stochastic process
 +
 
 +
$$
 +
\Delta  ^ {(} n) X ( t)  =  X ( t) -
 +
\left ( \begin{array}{c}
 +
n \\
 +
1
 +
\end{array}
 +
\right ) X ( t - 1 ) +
 +
\dots + ( - 1 )  ^ {n} \left ( \begin{array}{c}
 +
n \\
 +
n
 +
\end{array}
 +
\right )
 +
X ( t - n ) ,
 +
$$
 +
 
 +
where the  $  ( {} _ {k}  ^ {n} ) $
 +
are binomial coefficients. If  $  X ( t) $
 +
is a stochastic process with stationary increments of order  $  n $,
 +
then the process  $  \Delta  ^ {(} n) X ( t) $
 +
is stationary in the usual sense. Thus, in the case of discrete time, the theory of stochastic processes with stationary increments reduces easily to that of the more particular stationary stochastic processes. However, from the point of view of applications, the use of the concept of a stochastic process with stationary increments and discrete time  $  t $
 +
often turns out to be very convenient, since for many explicit non-stationary time series  $  x ( t) $,
 +
$  t = 1 , 2 \dots $
 +
met in practice, the series of their increments  $  \Delta  ^ {(} n) x ( t) $
 +
of some order  $  n $
 +
can be regarded as realizations of a stationary stochastic process  $  \Delta  ^ {(} n) X ( t) $.
 +
In particular, G. Box and G. Jenkins showed in [[#References|[1]]] that, when solving many practical problems, real time series can often be regarded as realizations of a so-called [[Auto-regressive process|auto-regressive process]], an integrated moving-average process that represents a special stochastic process with stationary increments and discrete time (see also [[#References|[2]]]–[[#References|[4]]]).
 +
 
 +
Examples of stochastic processes with stationary increments of the first order (in the strict sense) and in continuous time $  t $
 +
are a [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]]. Both of these also belong to the narrower class of processes with independent increments of the first order. In the case of continuous $  t $,  
 +
the theory of stochastic processes with stationary increments does not reduce directly to the theory of the simpler stationary processes. The correlation theory of stochastic processes with stationary increments of the first order (that is, the theory of the corresponding processes in the wide sense), was developed by A.N. Kolmogorov [[#References|[5]]] (see also [[#References|[6]]]). An analogous theory of stochastic processes with stationary increments of order $  n $,  
 +
where $  n $
 +
is an arbitrary positive integer, was considered in [[#References|[7]]]–[[#References|[9]]]. A central position in the correlation theory of stochastic processes with stationary increments is occupied by the derivation of the spectral decomposition of such processes and of their second-order moments. The concept of a generalized stochastic process (cf. [[Stochastic process, generalized|Stochastic process, generalized]]) can be used to simplify the theory of stochastic processes with stationary increments. Since in the theory of generalized stochastic processes, any stochastic process $  X ( t) $
 +
has derivatives of all orders (which are again generalized stochastic processes), a stochastic process with stationary increments of order $  n $
 +
can be defined as a stochastic process $  X ( t) $
 +
whose $  n $-
 +
th derivative $  X  ^ {(} n) $
 +
is an (in general, generalized) stationary stochastic process (see [[#References|[9]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1970) {{MR|0272138}} {{ZBL|0249.62009}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C.R. Nelson, "Applied time series analysis for managerial forecasting" , Holden-Day (1973) {{MR|}} {{ZBL|0271.62113}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O.D. Anderson, "Time series analysis and forecasting. The Box–Jenkins approach" , Butterworths (1976) {{MR|0448760}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.A. Robinson, M.T. Silva, "Digital foundations of time series analysis: The Box–Jenkins approach" , Holden-Day (1979)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.N. Kolmogorov, "Kurven im Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind" ''Dokl. Akad. Nauk SSSR'' , '''26''' : 1 (1940) pp. 6–9</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A.M. Yaglom, "Correlation theory of processes with stationary random increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025035.png" />" ''Transl. Amer. Math. Soc. (2)'' , '''8''' (1958) pp. 87–141 ''Mat. Sb.'' , '''37''' (1955) pp. 141–196</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M.S. Pinsker, "Theory of curves in Hilbert space with stationary increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025036.png" />" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''19''' (1955) pp. 319–345 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> K. Itô, "Stationary random distributions" ''Mem. Coll. Sci. Univ. Kyoto (A)'' , '''28''' (1954) pp. 209–223 {{MR|0065060}} {{ZBL|0059.11505}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1970) {{MR|0272138}} {{ZBL|0249.62009}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C.R. Nelson, "Applied time series analysis for managerial forecasting" , Holden-Day (1973) {{MR|}} {{ZBL|0271.62113}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O.D. Anderson, "Time series analysis and forecasting. The Box–Jenkins approach" , Butterworths (1976) {{MR|0448760}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.A. Robinson, M.T. Silva, "Digital foundations of time series analysis: The Box–Jenkins approach" , Holden-Day (1979)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.N. Kolmogorov, "Kurven im Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind" ''Dokl. Akad. Nauk SSSR'' , '''26''' : 1 (1940) pp. 6–9</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A.M. Yaglom, "Correlation theory of processes with stationary random increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025035.png" />" ''Transl. Amer. Math. Soc. (2)'' , '''8''' (1958) pp. 87–141 ''Mat. Sb.'' , '''37''' (1955) pp. 141–196</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M.S. Pinsker, "Theory of curves in Hilbert space with stationary increments of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090250/s09025036.png" />" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''19''' (1955) pp. 319–345 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> K. Itô, "Stationary random distributions" ''Mem. Coll. Sci. Univ. Kyoto (A)'' , '''28''' (1954) pp. 209–223 {{MR|0065060}} {{ZBL|0059.11505}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For additional references, see [[Stochastic process|Stochastic process]].
 
For additional references, see [[Stochastic process|Stochastic process]].

Latest revision as of 08:23, 6 June 2020


2010 Mathematics Subject Classification: Primary: 60G99 Secondary: 60G10 [MSN][ZBL]

A stochastic process $ X ( t) $ in discrete or continuous time $ t $ such that the statistical characteristics of its increments of some fixed order do not vary with time (that is, are invariant with respect to the time shifts $ t \mapsto t + a $). As in the case of stationary stochastic processes (cf. Stationary stochastic process), one distinguishes two types of such processes, namely stochastic processes with stationary increments in the strict sense, for which all finite-dimensional probability distributions of increments of $ X ( t) $ of a given order at the points $ t _ {1} \dots t _ {n} $ and $ t _ {1} + a \dots t _ {n} + a $ for any $ a $ coincide with one another, and stochastic processes with stationary increments in the wide sense, for which the mean values of an increment at $ t $ and the second moments of the increments at $ t $ and $ t+ s $ do not depend on $ t $.

In the case of processes $ X ( t) $ in discrete time $ t = 0 , \pm 1 \dots $ one can always pass from the consideration of $ X ( t) $ to that of the new stochastic process

$$ \Delta ^ {(} n) X ( t) = X ( t) - \left ( \begin{array}{c} n \\ 1 \end{array} \right ) X ( t - 1 ) + \dots + ( - 1 ) ^ {n} \left ( \begin{array}{c} n \\ n \end{array} \right ) X ( t - n ) , $$

where the $ ( {} _ {k} ^ {n} ) $ are binomial coefficients. If $ X ( t) $ is a stochastic process with stationary increments of order $ n $, then the process $ \Delta ^ {(} n) X ( t) $ is stationary in the usual sense. Thus, in the case of discrete time, the theory of stochastic processes with stationary increments reduces easily to that of the more particular stationary stochastic processes. However, from the point of view of applications, the use of the concept of a stochastic process with stationary increments and discrete time $ t $ often turns out to be very convenient, since for many explicit non-stationary time series $ x ( t) $, $ t = 1 , 2 \dots $ met in practice, the series of their increments $ \Delta ^ {(} n) x ( t) $ of some order $ n $ can be regarded as realizations of a stationary stochastic process $ \Delta ^ {(} n) X ( t) $. In particular, G. Box and G. Jenkins showed in [1] that, when solving many practical problems, real time series can often be regarded as realizations of a so-called auto-regressive process, an integrated moving-average process that represents a special stochastic process with stationary increments and discrete time (see also [2][4]).

Examples of stochastic processes with stationary increments of the first order (in the strict sense) and in continuous time $ t $ are a Wiener process and a Poisson process. Both of these also belong to the narrower class of processes with independent increments of the first order. In the case of continuous $ t $, the theory of stochastic processes with stationary increments does not reduce directly to the theory of the simpler stationary processes. The correlation theory of stochastic processes with stationary increments of the first order (that is, the theory of the corresponding processes in the wide sense), was developed by A.N. Kolmogorov [5] (see also [6]). An analogous theory of stochastic processes with stationary increments of order $ n $, where $ n $ is an arbitrary positive integer, was considered in [7][9]. A central position in the correlation theory of stochastic processes with stationary increments is occupied by the derivation of the spectral decomposition of such processes and of their second-order moments. The concept of a generalized stochastic process (cf. Stochastic process, generalized) can be used to simplify the theory of stochastic processes with stationary increments. Since in the theory of generalized stochastic processes, any stochastic process $ X ( t) $ has derivatives of all orders (which are again generalized stochastic processes), a stochastic process with stationary increments of order $ n $ can be defined as a stochastic process $ X ( t) $ whose $ n $- th derivative $ X ^ {(} n) $ is an (in general, generalized) stationary stochastic process (see [9]).

References

[1] G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1970) MR0272138 Zbl 0249.62009
[2] C.R. Nelson, "Applied time series analysis for managerial forecasting" , Holden-Day (1973) Zbl 0271.62113
[3] O.D. Anderson, "Time series analysis and forecasting. The Box–Jenkins approach" , Butterworths (1976) MR0448760
[4] E.A. Robinson, M.T. Silva, "Digital foundations of time series analysis: The Box–Jenkins approach" , Holden-Day (1979)
[5] A.N. Kolmogorov, "Kurven im Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind" Dokl. Akad. Nauk SSSR , 26 : 1 (1940) pp. 6–9
[6] J.L. Doob, "Stochastic processes" , Wiley (1953) MR1570654 MR0058896 Zbl 0053.26802
[7] A.M. Yaglom, "Correlation theory of processes with stationary random increments of order " Transl. Amer. Math. Soc. (2) , 8 (1958) pp. 87–141 Mat. Sb. , 37 (1955) pp. 141–196
[8] M.S. Pinsker, "Theory of curves in Hilbert space with stationary increments of order " Izv. Akad. Nauk SSSR Ser. Mat. , 19 (1955) pp. 319–345 (In Russian)
[9] K. Itô, "Stationary random distributions" Mem. Coll. Sci. Univ. Kyoto (A) , 28 (1954) pp. 209–223 MR0065060 Zbl 0059.11505

Comments

For additional references, see Stochastic process.

How to Cite This Entry:
Stochastic process with stationary increments. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_process_with_stationary_increments&oldid=23666
This article was adapted from an original article by A.M. Yaglom (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article