Smoothing of the characteristics of a random flow by random shifting of its parameters (Q1113555)

The random flow of homogeneous events with parameter \(\lambda_ 0(t)\) at time \(t\in (-\infty,+\infty)\) is studied. The events of the flow undergo shifting along the t-axis, and the shifts of the events are i.i.d. r.v. with the distribution function H, \(H(+\infty)=1\). The new flow formed by the shifted events of the original flow has the parameter \(\lambda_ 1(\cdot)\) given by the formula \[ (1)\quad \lambda_ 1(t)=\int^{t}_{-\infty}\lambda_ 0(u)dH(t-u). \] It is supposed that \(\lambda_ 0\) belongs to a class of T-periodic functions which admit a convergent Fourier series expansion on [0,T] and that \(\lambda_ 0\), \(\lambda_ 1\) belong to a normed space S with norm \(\| \cdot \|_ S\). Let \(| L|_ S\) be the norm of the linear filter (1) and \[ H_ S(\cdot)=\| \lambda_ 1(\cdot)-{\bar \lambda}\|_ S/\| \lambda_ 0(\cdot)-{\bar \lambda}\|_ S,\quad where\quad {\bar \lambda}=T^{-1}\int^{T}_{0}\lambda_ i(t)dt,\quad i=0,1. \] It is shown that for \(| L|_ S\leq 1\) for all spaces S (under consideration), \(H_ S(\cdot)\leq | L|_ S\). This expresses the effect of the ``smoothing'' by the random shifting.