Stochastic processes Flashcards
Stochastic process
Def: sequence of random variables A realization of the random process is a realization of different random variables
First order markov process
For all t, the conditional distribution of w_t depends only on the first lag f(w_t|w_t-1,w_t-2…)=f(w_t|w_t-1)
Strictly stationary process
For each set of indices t1, …, tn, the joint distribution of w_t1, w_t2, w_t3… depends only on the differences between the indices.
- Implications:
- marginal distributions are stationary
- moments of the w_t are time independent
- Covariances between any two variables only depends on their distance in the sequence
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/038/847/a_image_thumb.png?1456787367)
Covariance stationarity (weakly stationary)
Mean, and covariances are stationary.
- A covariance stationary normal process is stricly strationary
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/038/848/a_image_thumb.png?1456766873)
White noise process
Covariance stationary process with: - zero unconditional mean - constant unconditional variance - zero across-tome covariances If we add f(w_t|w_t-1,…,w1)=f(w_t) we have an independent white noise process.
Ergodic process
- A stationary process is ergodic if any two variables positioned far apart in the sequence are almost independently distributed.
- An ergodic process satisfies lim gamma_j=0 as j goes to infty
- Notice covariance stationarity do not imply ergodicity
Sufficient condition for ergodicity
- Covariance stationarity
- Series of covariances if absolutely convergent
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/040/266/a_image_thumb.png?1456767272)
LLN
We need:
- Constant mean
- Covariances depending only in differences of time indexes
- i.e. covariance stationarity
- Series of covariances converging absolutely
- With this two combined we have ergodicity
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/040/431/a_image_thumb.png?1455977566)
CLT
- Covariance stationarity
- Ergodicity
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/040/507/a_image_thumb.png?1455977736)
Normality in AR(1)
If we have:
- Conditional normality w_t|w_t-1 is normal
- Weak stationarity
- Normality of w_0
Then the whole process is jointly normal
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/040/552/a_image_thumb.png?1455978418)
Variance of sum of covariance stationary variables
![](https://s3.amazonaws.com/brainscape-prod/system/cm/181/409/071/a_image_thumb.png?1456246936)
Martigale difference sequence
![](https://s3.amazonaws.com/brainscape-prod/system/cm/182/009/607/a_image_thumb.png?1456766967)
Independent white noise
White noise +
![](https://s3.amazonaws.com/brainscape-prod/system/cm/182/009/695/a_image_thumb.png?1456767061)
Condition for applicability of CLT
w_t is an infinite-order moving average process
![](https://s3.amazonaws.com/brainscape-prod/system/cm/182/010/970/a_image_thumb.png?1456767643)
AR(1) definition
![](https://s3.amazonaws.com/brainscape-prod/system/cm/182/011/083/a_image_thumb.png?1456767741)