Figure 1

The eigenvalues ${\lambda }_q$ of the matrix $\mathbf{M}\mathbf{C}$ for the time-averaged MSD of Brownian motion, with $N=200$. The vertical axis is normalized by $2{\kappa }_1={\lambda }_1+\cdots +{\lambda }_N=2n$ so that ${\lambda }_q/\left(2{\kappa }_1\right)$ can be seen as a relative weight of each eigenvalue. (a) ${\lambda }_q$ as a function of $q$ for six lag times $n=1,10,50,100,150,190$. (b) Three largest eigenvalues ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ as functions of the lag time $n$. For large $n$, the first eigenvalue ${\lambda }_1$ provides the overwhelming contribution.Reuse & Permissions

Figure 2

The probability density $p\left(z\right)$ (lines) and its approximations (symbols) by the generalized Gamma distribution, for the time-averaged MSD of Brownian motion ($H=0.5$), with $N=200$ and three lag times: $n=50$ (blue solid curve, circles and pluses), $n=100$ (green dashes line, squares and crosses), and $n=150$ (red dash-dotted line, triangles and asterisks), shown at semilogarithmic (a) and logarithmic (b) scales. The horizontal axis is normalized by the mean value ${\kappa }_1=n$ (the solid vertical line points on the normalized mean value at 1). Circles, squares, and triangles show the first approximation by fitting $p\left(z\right)$ to Eq. (17). Three dotted vertical lines indicate the most probable value of this approximation according to Eq. (20). Pluses, crosses, and asterisks indicate the second approximation obtained by computing three moments, as discussed in Sec. 2e.Reuse & Permissions

Figure 3

The eigenvalues ${\lambda }_q$ of the matrix $\mathbf{M}\mathbf{C}$ for the time-averaged MSD of fBm, with $N=200$, six lag times $n=1,10,50,100,150,190$, and $H=0.75$ (a) and $H=0.25$ (b). The eigenvalues ${\lambda }_q$ are normalized by $2{\kappa }_1=2n^{2H}$.Reuse & Permissions

Figure 4

The probability density $p\left(z\right)$ (lines) and its approximations (symbols) by a generalized Gamma distribution, for the time-averaged MSD of fBm, with $H=0.25$, $N=200$ and three lag times: $n=50$ (blue solid curve, circles and pluses), $n=100$ (green dashes line, squares and crosses), and $n=150$ (red dash-dotted line, triangles and asterisks), shown at semilogarithmic (a) and logarithmic (b) scales. The horizontal axis is normalized by the mean value ${\kappa }_1=n^{2H}$ (the solid vertical line points on the normalized mean value at 1). Circles, squares, and triangles show the first approximation by fitting $p\left(z\right)$ to Eq. (17). Three dotted vertical lines indicate the most probable value of this approximation according to Eq. (20). Pluses, crosses, and asterisks indicate the second approximation obtained by computing three moments, as discussed in Sec. 2e.Reuse & Permissions

Figure 5

The probability density $p\left(z\right)$ (lines) and its approximations (symbols) by a generalized Gamma distribution, for the time-averaged MSD of fBm, with $H=0.75$, $N=200$ and three lag times: $n=50$ (blue solid curve, circles and pluses), $n=100$ (green dashes line, squares and crosses), and $n=150$ (red dash-dotted line, triangles and asterisks), shown at semilogarithmic (a) and logarithmic (b) scales. The horizontal axis is normalized by the mean value ${\kappa }_1=n^{2H}$ (the solid vertical line points on the normalized mean value at 1). Circles, squares, and triangles show the first approximation by fitting $p\left(z\right)$ to Eq. (17). Three dotted vertical lines indicate the most probable value of this approximation according to Eq. (20). Pluses, crosses, and asterisks indicate the second approximation obtained by computing three moments, as discussed in Sec. 2e.Reuse & Permissions

Figure 6

The reduced variance ${\kappa }_2/{\kappa }_1^2$ (a), the skewness ${\kappa }_3/{\kappa }_2^{3/2}$ (b), and the kurtosis ${\kappa }_4/{\kappa }_2^2$ (c) of the time-averaged MSD as functions of the lag time $n$, with different $H$.Reuse & Permissions

Figure 7

The probability density $p\left(z\right)$ (solid line) of the time-averaged MSD of Brownian motion, with $N=10$ and $n=3$, shown at semilogarithmic (a) and logarithmic (b) scales. For comparison, we present the best fit to Eq. (17) (circles), the expansion (A2) over Laguerre polynomials (dashed line), the expansion (A3) over chi-squared distributions (pluses), and the power series expansion (A1) (crosses). These three expansions are truncated to $K=10$ (11 terms).Reuse & Permissions

Figure 8

Correlation $\rho$ between the time-averaged MSD with the lag time 1 (${\chi }_1$) and the time-averaged MSD with the lag time $n$ (${\chi }_2$), for persistent ($H=0.75$), normal ($H=0.5$), and antipersistent ($H=0.25$) fBm.Reuse & Permissions