Let us consider a sequence of i.i.d. (independent and identically distributed) variables over time, \(X_{t}, t=1 \quad H\). Let and be the expected value and standard deviation, respectively, for each variable \(X_{t}\). If we sum variables over the \(H\) time periods, we define a new variable,

\[Y={ }_{t=1}^{H} X_{t}\]

and find \[\begin{equation*}
\mathrm{E}[Y]={ }_{t=1}^{H} \mathrm{E}\left[X_{t}\right]=H \tag{2.12}
\end{equation*}\]
and \[\begin{equation*}
Y=\bar{H} \operatorname{Var}\left(X_{t}\right)=\bar{H} \tag{2.13}
\end{equation*}\]
Note that we are writing the variance of a sum as a sum of variances, which is wrong in general, but it is fine under the assumption of independence (actually, lack of correlation is enough). Thus, we find that the expected value scales linearly with time, whereas the standard deviation scales with the square root of time. The two functions are compared in Fig. 2.4. We may observe that, for a large \(H\), the expected value dominates, but roles are reversed for a small \(H\) going to zero. We shall refer to this fact as the square-root rule.
This seems to suggest that, for a small time period, volatility does dominate drift, but the conclusion is not really warranted. To begin with, random returns should be multiplied, rather than summed. However, if we consider summing log-returns, or an approximation valid for small daily returns, we may claim that the reasoning makes sense. The assumption of independence of returns over time is bit more critical, but there are models based on this idea. Probably, the most delicate point is that we should consider "slicing" a random variable, i.e., expressing a given \(Y\) as a sum of terms \(X_{t}\), for a small time interval \(t\). If we add independent (uncorrelated) normals, we do get a normal. But can we slice a normal into smaller normals? We should resort to a more sophisticated concept related to self-similar stochastic processes. Indeed, we will use such a process, the Wiener process, based on a normal distribution. We cannot come up with a rigorous analysis here, but empirical data do suggest the idea that, on a short time interval of length \(t\), drift goes to zero more rapidly than volatility does. Put another way, we cannot reject the null hypothesis that return over \(t\) is zero.

Data From Fig. 2.4