Process with steady growth

The process with stationary increments , also known as the process with stationary increments , is a term from the theory of stochastic processes , a branch of probability theory . A process clearly has steady growth if the change in the process in a fixed time step does not change in the course of the development of the process. Examples of processes with steady growth are the Lévy process and thus also the Poisson and Wiener processes .

definition

A real-valued stochastic process with an index set that is closed with regard to addition is called a process with stationary increases if and only if the distribution of the random variables is for any ${\displaystyle (X_{t})_{t\in T}}$ ${\displaystyle T\subset \mathbb {R} }$ ${\displaystyle p,q,r\in T}$

${\displaystyle Y_{1}=X_{p+q+r}-X_{q+r}}$

with the distribution of the random variables

${\displaystyle Y_{2}=X_{p+r}-X_{r}}$

matches. Is it enough to put it. ${\displaystyle 0\in T}$ ${\displaystyle r=0}$

example

Consider as an example the symmetric random walk on , so by the stochastic process, which is defined ${\displaystyle \mathbb {Z} }$

${\displaystyle X_{0}=0}$

and ${\displaystyle X_{n}=\sum _{i=1}^{n}Z_{i}}$

for , where the independent Rademacher-distributed random variables are. So it applies . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle Z_{i}}$ ${\displaystyle P(Z_{n}=-1)=P(Z_{n}=1)={\tfrac {1}{2}}}$

Because of that , it is enough to bet. It follows ${\displaystyle T=\mathbb {N} }$ ${\displaystyle 0\in T}$ ${\displaystyle r=0}$

${\displaystyle Y_{1}=\sum _{i=1}^{p+q}Z_{i}-\sum _{i=1}^{q}Z_{i}=\sum _{i=q+1}^{p+q}Z_{i}}$

and

${\displaystyle Y_{2}=\sum _{i=1}^{p}Z_{i}-X_{0}}$.

Both and are therefore the sum of independent Rademacher-distributed random variables and thus have the same distribution. So the symmetrical odyssey is on a process with stationary increases. ${\displaystyle Y_{1}}$ ${\displaystyle Y_{2}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} }$

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 .