Stochastic kernel

A stochastic kernel is the transition function of a (usually discrete-time) stochastic process. Often, it is assumed to be iid, thus a probability density function.

Formally a density can be


 * $$f_{\lambda}(y)=\frac{1}{I\lambda}\sum_{i=1}^{I}K\left(\frac{y-y_{i}}{\lambda}\right),$$

where $$y_{i}$$ is the observed series, $$\lambda$$ is the bandwidth, and K is the kernel function.

Examples

 * The uniform kernel is $$K=1/2$$ for $$-1<t<1$$.
 * The triangular kernel is $$K=1-|t|$$ for $$-1<t<1 $$.
 * The quartic kernel is $$K=(15/16)(1-t^2)^2$$ for $$-1<t<1$$.
 * The Epanechnikov kernel is $$K=(3/4)(1-t^2)$$ for $$-1<t<1$$.

Übergangswahrscheinlichkeit Probabilità di transizione