Annualization of discrete returns

There is a well known approach to annualize volatility of log-returns for a given frequency. Let $P(t)$ a price process and define a log return $r_l(t)$ as $r_l(t) = \ln \left( \frac{P(t)}{P(t-1)} \right).$ An aggregate return over $n$ periods is \begin{equation} \begin{aligned} r_l^A(t) &= \ln \left( \frac{P(t)}{P(t-n)} \right) \\ &= \sum_{t-n+1}^t r_l(i). \end{aligned} \end{equation} Let $\sigma_l^2$ denote variance of log-returns. Then, variance of an aggregate return is $$ \