Some remarks on integral parameters of Wiener process

Abstract

We consider integrals $\tau_{\rho}=\int_0^1\rho\xi^2\,dx$, where $\xi$ is Wiener process and $\rho$ is generalized function from some class of multipliers. In the case when multiplier $\rho$ belongs to the trace-class, it is shown that $\tau_{\rho}$ has $\chi^2$-distribution (or analogous). An example of multiplier $\rho$ not belonging to the trace-class is constructed.