Canonical stochastic process

A canonical stochastic process , canonical process for short , is a general formulation of a stochastic process in probability theory , which is characterized by its simplicity. The coordinate mappings of a large underlying probability space are understood as random variables of the stochastic process. The underlying measurement space is then also referred to as the canonical space .

definition

An arbitrary non-empty index set as well as a non-empty basic set and a σ-algebra on this basic set are given. Looking at the projections ${\displaystyle I}$ ${\displaystyle E}$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle \pi _{i}:(E^{I},{\mathcal {E}}^{\otimes I})\to (E,{\mathcal {E}})}$,

which are defined for all by ${\displaystyle i\in I}$

${\displaystyle \pi _{i}(\omega )=\omega _{i}}$,

Sun is the stochastic process of the canonical process on . The measuring space is then also called the canonical space of the process. ${\displaystyle (X_{i})_{i\in I}:=(\pi _{i})_{i\in I}}$ ${\displaystyle (E^{I},{\mathcal {E}}^{\otimes I})}$ ${\displaystyle (E^{I},{\mathcal {E}}^{\otimes I})}$

comment

The distributions of the random variables are defined by specifying a probability measure in the measurement space ; they are then exactly the one-dimensional marginal distributions . For this one might need statements about the existence of probability measures on countable or uncountable products of sets like the Ionescu-Tulcea theorem or Kolmogorov's extension theorem . ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle P}$ ${\displaystyle (E^{I},{\mathcal {E}}^{\otimes I})}$

example

If one considers the index set as well as the basic space provided with Borel's σ-algebra , i.e. and any probability measure on and the product measure , then the projections on the individual components have the distributions . The canonical process provides independent, identically distributed random variables due to the properties of the product dimension . ${\displaystyle I=\mathbb {N} }$ ${\displaystyle E=\mathbb {R} }$ ${\displaystyle {\mathcal {E}}={\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle {\tilde {P}}}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle P={\tilde {P}}^{\otimes \mathbb {N} }}$ ${\displaystyle P_{\pi _{i}}={\tilde {P}}}$ ${\displaystyle {\tilde {P}}}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 281 , doi : 10.1007 / 978-3-642-36018-3 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 174 , doi : 10.1007 / 978-3-642-45387-8 .