On a Characterization of Brownian Motion

Abstract

Let $X(t)$ be a homogeneous random process with independent increments, and $a(t)$, $b(t)$ certain functions that are continuous in the closed interval a, b. Then the following theorem is true:If\[ \int_a^b {[a(t)b(t)]^2 dt \ne 0,} \] the integrals\[ Y = \int_a^b {a(t)dX(t)} {\textit{ and }}Z\int_a^b {b(t)dX(t)} \]are independent, and if at least one of the integrals\[ \int_a^b {\frac{{a^2 (t)}} {{b^2 (t)}}dt\,} ,\int_a^b {\frac{{b^2 (t)}} {{a^2 (t)}}dt\,} \]exists, then the process$X(t)$is a Brownian motion.A similar theorem is proved for infinite linear forms\[ \sum\limits_{k = 1}^\infty {a_k X_k } ,\sum\limits_{k = 1}^\infty {b_k X_k } .\]