VO Stochastic Processes - Zusammenfassung Prüfung
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General
Given a probability space a set called time space and a set called state space. A stochastic process is a function so that is jointly measurable, is a random variable from into and is a measurable function called trajectory.
If is denumerable, the process is called discrete-state process, otherwise continuous-state process. If is denumerable one speaks of a process in discrete-time, otherwise of a process in continuous-time.
Discrete
A discrete-event process takes only a finite number of values in bounded time. A discrete-event process can be represented by a sequence and a process in discrete-time. are called jump times, are called sojourn times and the process in discrete-time is called embedded process.
Stationary & Homogenous
A process is strongly stationary if . A process is weakly stationary if it has constant expectation and variance over time. A process is homogeneous if it has stationary increments so that does not depend on . The expected visits of a process started in state to state is In matrix notation Let be the first return time .
State Properties
A state can have the following state-properties.
- reachable: (written ), if .
- commuting: (written ), if .
- recurrent: if
- positive recurrent: if .
- null recurrent: if
- transient: if .
- period : if is denumerable, is recurrent and .
- periodic: if has period .
- aperiodic: if does not have period ..
- absorbing: if .
defines partial-ordered equivalence classes on A class precedes if . All states in a class possess the same state-property. A class has a class-properties if its states possess the corresponding state-property. A class is called maximal if it precedes no other class. A chain with only one class is irreducible, otherwise reducible. A positive recurrent, irreducible chain with period 1 is called ergodic. A process is called independent if . A random variable is called defective, it its distribution function does not converge to 1, so that is transient if is defective. A distribution is stationary if , with if is transient.
Process Properties
The Chapman-Kolmogorov equation is a statement of the formula of the total probability modified by the Markov Property. It connects the distribution of a process at with through an intermediate time , by . For homogeneous processes this implies in discrete-time and in continuous-time. It is necessary but not sufficient for the Markov property, . A discrete-state Markov process is called Markov chain. A process has the Martingale property if .
Ergodic Coefficient
Define Dubrushin's coefficient of ergodicity as
with properties
- .
- .
- the smaller the more independent (ergodic) the process. has identical rows.
- is sub-multiplicative.
The following are equivalent
- is ergodic.
- has a unique stationary distribution.
- .
- . \quad (LLN)
- exponentially fast. \quad (CLT)
- (Ergodic Theorem)
Continuous Time
By the Chapman-Kolmogorov equation, a homogeneous Markov chain in continuous time is determined by a semi-group of transition matrices. A semi-group is continuous if and equi-continuous if
If is [equi-]continuous, than is uniformly [equi-]continuous. is a sub-additative function.
For a small time interval , and , so that the intensity matrix has the form
- for all and a constant .
- for .
- for all .
with properties
- .
- (element-wise).
- is absorbing.
- The sojourn time at state is exponentially
distributed with rate . Conditioned on leaving to state the sojourn time is exponential distributed with rate .
The embedded process has transition matrix with
is stationary for for all where is the stationary distribution of .
Death / Birth Process
A Markov chain in continuous time is called birth/death process if and
A Poisson process is a birth/death process with and . A queue is a birth/death process with and . Its embedded process is
- positive recurrent for .
- negative recurrent for .
- transient for .
The queue has the following properties.
- expected length of idle period.
- expected length of busy period.
- expected number of costumers in busy period.
- probability of being immediately severed.
- expected length of the queue.
- expected waiting time for not immediately served customers.
- expected overall waiting time.
Continuous Time & States
Assume that for a certain person the observed time till this persons death is the realization of a random variable with density and distribution function Then the survivor function, is the probability that the survival time for this person is greater or equal to , so that Also define the hazard function as probability that an individual dies at time , given that the person has survived up to time
and the cumulative hazard function as Now using the definition of the condition probability and can be written as and
If than one defines a Wiener process , starting from a random walk with step-size and step-frequency with steps in time span , by setting and letting , so that . Define the Ornstein-Uhlenbeck process and Brownian bridge as and .
and Gaußprocess with expectation 0 and covariance , and .
- and are Markov processes.
- Only has Martingale property.
- Only has identical distributed increments.
- Only has dependent increments.
- Only is stationary.
- The loglikelihood-ratio test-static is asympothically .
- The Kolmogorov-Smirnov test is asymptoically .