VO Stochastic Processes - Zusammenfassung Buch
is a collection of random variables defined on some probability space
|t||fixed||is a random variable|
|w||fixed||is a function on (trajectory or path of the stoch. process)|
|t||discrete||discrete time porcesses, most import ones are the Markov processes|
Ergodicity allows us to extract all the information we need from 1 trajectory because
because something like LLN is valid.
The lower the ergodicity coefficient the faster is convergence
- The process is called a homogenous Markov Chain if there exists
- The probablility KLAUS of a homogenous Markov Chain is determined by the distribution of (starting distribution) and the transition matrix .
- If the process is startetd with a starting distribution and
then the distribution of is
- Expected number of visists in state if started in is:
- Putting values together in matrix
- is reachable from if there exists with :
- Commuting states:
- Let be the equivalnce classes of commuting states. We introduce a partial ordering for these classes by saying that
- preceedes (in symbol ) if for all and all
(there may also exist incomparable classes)
- A class is called transient, if it preceeds another class. Otherwise the class is called a maximal class.
- A state is called recurrent if the chain which is started in returns with probability 1 to .