Convolutional semigroup

In probability theory, a convolution half-group is a family of probability measures that is in a certain sense stable with respect to convolution . Convolution half-groups occur, for example, in the investigation of characteristic functions or as an aid to the construction of stochastic processes with certain properties, such as the Wiener process .

definition

A semigroup with regard to the connection and a family of probability measures are given . Let it denote the convolution of and . ${\displaystyle I\subset [0,+\infty )}$ ${\displaystyle +}$ ${\displaystyle (\mu _{t})_{t\in I}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \nu _{1}*\nu _{2}}$ ${\displaystyle \nu _{1}}$ ${\displaystyle \nu _{2}}$

The family is now called a convolution half-group , if for all ${\displaystyle (\mu _{t})_{t\in I}}$ ${\displaystyle r,s\in I}$

${\displaystyle \mu _{r}*\mu _{s}=\mu _{r+s}}$

applies.

Examples

The following examples can be justified by means of characteristic functions . For this purpose, one takes advantage of the fact that the convolution of the probability measures of the distribution corresponds to the sum of the random variables and this in turn is described by the product of the characteristic function.

• Normal distribution : The normal distribution is a convolution half-group in both parameters , because it applies to all and . Thus, for fixes there is always a folding half group just as there is for fixes a folding half group . ${\displaystyle \mu ,\sigma ^{2}}$ ${\displaystyle {\mathcal {N}}_{\mu _{1},\sigma _{1}^{2}}*{\mathcal {N}}_{\mu _{2},\sigma _{2}^{2}}={\mathcal {N}}_{\mu _{1}+\mu _{2},\sigma _{1}^{2}+\sigma _{2}^{2}}}$ ${\displaystyle \mu _{1},\mu _{2}\in \mathbb {R} }$ ${\displaystyle \sigma _{1}^{2},\sigma _{2}^{2}>0}$ ${\displaystyle \mu }$ ${\displaystyle ({\mathcal {N}}_{\mu ,\sigma ^{2}})_{\sigma ^{2}>0}}$ ${\displaystyle ({\mathcal {N}}_{\mu ,\sigma ^{2}})_{\mu \geq 0}}$ ${\displaystyle \sigma ^{2}}$
• Gamma distribution : The gamma distribution is two-parameter, but only forms a convolution half-group in the second parameter, because it is for fixed and always . ${\displaystyle \vartheta }$ ${\displaystyle r,s>0}$ ${\displaystyle \Gamma _{\vartheta ,r}*\Gamma _{\vartheta ,s}=\Gamma _{\vartheta ,r+s}}$
• Further folding half groups with the half group form the Cauchy distribution , the Dirac distribution and the Poisson distribution . Examples of convolution semigroups with respect to the semigroup are the binomial distribution , the Erlang distribution , the chi-square distribution and the negative binomial distribution . ${\displaystyle ((0,\infty ),+)}$ ${\displaystyle (\mathbb {N} ,+)}$

Tightening

Continuous folding semi-group

A convolution half-group is called a continuous convolution half-group with respect to the weak convergence , if is and holds. Here denotes the Dirac measure on the 0th ${\displaystyle (\mu _{t})_{t\in I}}$ ${\displaystyle I=[0,\infty )}$ ${\displaystyle \lim _{t\to 0}\mu _{t}=\delta _{0}}$ ${\displaystyle \delta _{0}}$

Non-negative convolutional semigroup

A convolution semigroup of probability measures on is a non-negative convolution semigroup if for ever is. ${\displaystyle (\mu _{t})_{t\in I}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle t\in I}$ ${\displaystyle \mu _{t}((-\infty ,0))=0}$

properties

Cores by convolutional semigroups

Markov nuclei can be defined by folding semigroups , which form a transition semigroup. To do this, one defines and ${\displaystyle \mu _{0}=\delta _{0}}$

${\displaystyle K_{t}(x,\mathrm {d} y):=\delta _{x}*\mu _{t}(\mathrm {d} y)}$.

Then the Chapman-Kolmogorow equation applies , because with the calculation rules for the convolution and concatenation of kernels follows

${\displaystyle K_{s}\cdot K_{t}=K_{s+t}}$.

Like any transition semigroup, the kernels also define a consistent family of stochastic kernels .

Stochastic processes through convolution half-groups

Convolution half-groups can also be used to define stochastic processes that have independent gains and stationary gains . Conversely, every stochastic process with independent stationary increments defines a convolution half-group. The best-known example here is the Wiener process , which can be constructed from the convolution half-group except for the continuity of its paths . In doing so, one takes advantage of the fact that every consistent family of stochastic kernels with an index set for a given probability measure is defined on a unique probability measure . Thus, the conclusion follows from the convolution half group to the transition half group to the consistent family for the uniqueness of the probability measure with the required properties. ${\displaystyle ({\mathcal {N}}_{0,t})_{t\geq 0}}$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle \nu }$ ${\displaystyle E}$ ${\displaystyle (E^{I},{\mathcal {B}}(E)^{\otimes I})}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 297-300 , doi : 10.1007 / 978-3-642-36018-3 .