The statistics of the digits of a continued fraction, also known as partial quotients, have been studied at least since the time of Gauss. The usual measure $m$ on the open interval $(0,1)$ gives a probability space ${\mathcal U}$. Let $a_k$, $k\ge 1$ be integer-valued random variables which take $\alpha\in (0,1)$ to the $k^{{\rm th}}$ partial quotient or digit in the continued fraction expansion $\alpha=1/(a_1+1/(a_2+\cdots))$. Let $S_r=S_r(\alpha)=\sum_{k=1}^r a_k$. It is well known that although there is an average value for $\log a_k$, each $a_k$, let alone each $S_r$, has infinite expected value or first moment.