# Stochastic kernel

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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

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

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

## Examples

• The uniform kernel is ${\displaystyle K=1/2}$ for ${\displaystyle -1.
• The triangular kernel is ${\displaystyle K=1-|t|}$ for ${\displaystyle -1.
• The quartic kernel is ${\displaystyle K=(15/16)(1-t^{2})^{2}}$ for ${\displaystyle -1.
• The Epanechnikov kernel is ${\displaystyle K=(3/4)(1-t^{2})}$ for ${\displaystyle -1.