Comment on “Temporal Correlations of the Running Maximum of a Brownian Trajectory”

Figure 1

(a) Histograms of estimates obtained from application of, respectively, ${\widehat{D}}_{\mathrm{msd}}^{(N)}$ (red) and ${\widehat{D}}_{\mathrm{es}}^{(N,k)}$ with $k=1$ (blue) to $10^5$ MC simulated, discretely sampled BM trajectories using $D=0.25$, $\mathrm{\Delta }t=1$, and $N=10^4$. (b) The same as (a) for $k=100$. (c) Mean values of estimates obtained as in (a) for various values of $N$. Results are shown for ${\widehat{D}}_{\mathrm{msd}}^{(N)}$ (pluses) and ${\widehat{D}}_{\mathrm{es}}^{(N,k)}$ with, respectively, $k$ values of 1 (full circles), 10 (open circles), and 100 (crosses). (d) The same as (c) for the variances of the estimates. The theoretical variance $2D^2/(N-1)$ for ${\widehat{D}}_{\mathrm{msd}}^{(N)}$, the Cramér-Rao lower bound, is indicated (full line).