Median, Mean, and Variance Stability of a Process Under Temporally Correlated Stochastic Feedback

Stochasticity in feedback gains leads to heavy-tailed state distributions in which the median, mean, and variance have different stability properties. These properties are characterised for a scalar system with temporally correlated stochastic feedback. Necessary and sufficient conditions are obtained for the generic case of log-normal feedback gain distributions. Temporal correlation is modeled by a multivariable Gaussian distribution of the logarithm of the feedback gain values. This correlation has no effect on the stability of the median of the state, but does influence the stability of both the mean and the variance of the state. Examples illustrate both stabilising and destabilising correlations.