Covariance function

The covariance is in the theory of stochastic processes , a branch of probability theory , a specific real-valued function which is associated with a stochastic process. The covariance function is important because a certain class of stochastic processes can be clearly characterized by its covariance function. Covariance functions are often found in the context of the Wiener process and related constructions.

definition

A stochastic process with an index set is given . Then the function is called ${\displaystyle X=(X_{t})_{t\in I}}$ ${\displaystyle I}$

${\displaystyle \Gamma \colon I\times I\to \mathbb {R} }$

defined by

${\displaystyle \Gamma (s,t)=\operatorname {Cov} (X_{s},X_{t}):=\mathbb {E} \left[(X_{s}-\mathbb {E} (X_{s}))\cdot (X_{t}-\mathbb {E} (X_{t}))\right]}$

the covariance function of the stochastic process . The covariance of the random variables denotes and . is the expected value. ${\displaystyle X}$ ${\displaystyle \operatorname {Cov} (X,Y)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \mathbb {E} }$

example

A Viennese trial is given . Is oBdA , so is ${\displaystyle X=(W_{t})_{t\geq 0}}$ ${\displaystyle 0\leq s<t}$

${\displaystyle \operatorname {Cov} (X_{s},X_{t})=\operatorname {Cov} (X_{s},X_{t}-X_{s}+X_{s})=\operatorname {Cov} (X_{s},X_{t}-X_{s})+\operatorname {Cov} (X_{s},X_{s})}$

Since the Vienna Process is a process with independent increments , and therefore applies ${\displaystyle \operatorname {Cov} (X_{s},X_{t}-X_{s})=0}$

${\displaystyle \operatorname {Cov} (X_{s},X_{t})=\operatorname {Cov} (X_{s},X_{s})=\operatorname {Var} (X_{s})=s}$

since the process has normally distributed increases. Thus applies to the Wiener process

${\displaystyle \Gamma (s,t)=\min(s,t)}$.

properties

Every Gaussian process that is centered in the sense that applies to all is uniquely determined by its covariance function. For there are given, then the distribution of the process can at these times as determined as follows: Since the process is normally distributed multi-dimensionally at these times and a multidimensional normal distribution uniquely identified by their expected value vector and the covariance matrix is determined, simply because of centeredness to determine the covariance matrix . But this is given by the covariance function: The entry in the i-th row and the j-th column is exact . ${\displaystyle X=(X_{t})_{t\in I}}$ ${\displaystyle E(X_{t})=0}$ ${\displaystyle t\in I}$ ${\displaystyle t_{0},t_{1},\dots ,t_{N}\in I}$ ${\displaystyle \Gamma (t_{i},t_{j})}$

This procedure can be carried out by any and all . The distributions obtained in this way then form a projective family and thus uniquely determine the distribution of the process according to Kolmogorov's extension theorem . ${\displaystyle t_{0},t_{1},\dots ,t_{N}\in I}$ ${\displaystyle N\in \mathbb {N} }$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .