Transformations of random processes
Random Processes - Basis and applications

A transformation of a random process is defined by the same function applied to any realization of the process, with the result Y(s_Y, ω) = f[X(s_X, ω)], where s_X and s_Y are the supports of X and Y respectively. The result of the transformation is itself a random process. In general, the function f can itself be random, for example in so-called "stochastic" algorithms. But in the context of this article, the function itself will be deterministic, which entails that the realization Y(s_Y, ω) is entirely determined by the realization X(s_X, ω). The function f may itself depend on the support, e.g. if s_y = s_x = t, we can write Y(t, ω) = f_t[X(t, ω)]. An averaging of X over a duration T, calculated at time t, will then be written $Y\left(t,\omega \right)=1\text{/}T$