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.

8 pages