It is known that if we have a stationary and ergodic process $\xi_1,\xi_2,\cdots$ with expected value 0 with probability 1 and $E(\xi^2)=v^2$ is positive and finite.
The standardized partial sum $S_n(p)=\frac{R_{[np]}}{\sqrt(n)v}$ where $R_n=\xi_1+\cdots+\xi_n$; then $S_n \longrightarrow W(p)$ as $n \longrightarrow $ infinity. $W(p)$ is a Brownian motion process.
Is it possible to prove a similar result if instead of $\xi_1,\xi_2,\cdots$ we have $c_1 \xi_1,c_2 \xi_2,\cdots$ where $c_1,c_2,\cdots$ are constants?