time $t$ by such a fluctuation occurring at time $t_1$ reads Y\_t=Y\_t-|Y\_t =(1+C ). \[eq:shift\] Note that in the asymptotic limit of interest, at first glance, the nontrivial term in this expression seems to be of order $1/t_1^{3/2}$, thus, if $t_1\sim t_0$, it is smaller than our accuracy goal. However, it is enhanced by the $e^{\delta}$ factor, which turns out to be large.

Probability distribution and moments
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We may convert the conjectured probability of a forward fluctuation of size $\delta$ (Eq. (\[eq:proba\])) into the probability distribution of the difference of $\ln Y$ between two times $t_0$ and $t>t_0$ by simple changes of variables. We first discuss the variable fY\_t-Y\_[t\_0]{}. The fundamental observation is that a fluctuation may essentially have two opposite effects on $\delta f\equiv \delta\ln Y_t-\delta\ln Y_{t_0}$. If it occurs after time $t_0$, then it gives a positive contribution. If instead it occurs before $t_0$, it generates a negative $\delta f$. Now we observe that the difference between $\delta f$ and $f$ reads $\ln {\bar Y_t}/{\bar Y_{t_0}}$, which is of order $1/\sqrt{t},1/\sqrt{t_0}$ and thus, we may trade $\delta f$ for $f$ (see Appendix \[sec:details\] for more details).\
Let us first address the case in which the fluctuation occurs 