1-Step Ahead Return Decomposition
First $r_t$ is a one period return. We can decompose $r_t$ as
$$ r_t = \mu_{t-1} + \epsilon_t $$
$\epsilon_t$ is called the 'noise component' and is the unpredictable component.
The predictable component as of time $t - 1$ is called $\mu_{t - 1}$.
$\mu_{t -1}$ is your best guess about $r_t$, considering all available information as of time $t - 1$.
$F_{t - 1}$ is called the information set as on $t - 1$.
$$ \mu_{t-1} := E[r_t|F_{t-1}] $$
$$ \epsilon_t := r_t - \mu_{t-1} $$
Different ways to predict returns.
'linear factor models' (CAPM, Fama-French 3 or 5 Factor models)
'Time-series Models' (ARMA (p,q))
If the assumption, that $\epsilon_t$ is a White Noise Process, not hold, we are going to use the ARCH(m), GARCH(m,s) Frameworks. To capture skewness in returns we extend these Models to the T-GARCH and E-GARCH.
ADD: The conditional one-step ahead return density of $r_t$, depends on the probability density that
you assume for $\epsilon_t$. For example, the one-step ahead density for $r_t$ is Gaussian if $\epsilon_t$ is Gaussian, and so on.
Multi-Step Ahead Return Decomposition
Let $j$ be the holding length, $0 < j < \infty$. Now, every $j$-period holding period return can be decomposed as
$$ r_{t,t+j} = \mu_{t,t+j} + \epsilon_{t,t+j} $$