On rational control of the mean level of random noise (Q1881871)

The author considers a sequence of random variables \(\xi_n,n=0,1,\dots\). The distribution of \(\xi_n\) depends on \(n\) and on a chosen control. There are two alternatives for the controls. These random variables are interpreted as noise. The aim of the control is to reduce the mean value of the noise. The proposed control algorithm is as follows. The random walk is defined starting at \(\eta_0=j_0\): if \(|\eta_n+\xi_n|\leq R\), then \(\eta_{n+1}=\eta_n+\xi_n\), otherwise \(\eta_{n+1}=\eta_n,n=0,1,\dots\). At time \(n\) one of two alternatives is chosen according to the rule: the first alternative is chosen if \(\eta_n>0\), the second alternative is chosen if \(\eta_n<0\), and both alternatives may be chosen if \(\eta_n=0\).