Fractional Edgeworth expansion: Corrections to the Gaussian-Lévy central-limit theorem

In this article we generalize the classical Edgeworth expansion for the probability density function (PDF) of sums of a finite number of symmetric independent identically distributed random variables with a finite variance to PDFs with a diverging variance, which converge to a Lévy $\alpha$-stable density function. Our correction may be written by means of a series of fractional derivatives of the Lévy and the conjugate Lévy PDFs. This series expansion is general and applies also to the Gaussian regime. To describe the terms in the series expansion, we introduce a new family of special functions and briefly discuss their properties. We implement our generalization to the distribution of the momentum for atoms undergoing Sisyphus cooling, and show the improvement of our leading order approximation compared to previous approximations. In vicinity of the transition between Lévy and Gauss behaviors, convergence to asymptotic results slows down.

Figure 1

$T_{\alpha ,\gamma }\left(\tilde{x}\right)$ for $\alpha =0.5$, and for various $\gamma$ values. The $T_{\alpha ,\gamma }\left(\tilde{x}\right)$ values were calculated both by numerical inverse Fourier transform of the expression in Eq. (17) (markers) and by calculating the series expansion in Eq. (26) with $2\times {10}^5$ terms (dashed lines). Note that we only illustrate the regime $\gamma >\alpha$, since these are the terms that appear in our series expansion.

Figure 2

$T_{\alpha ,\gamma }\left(\tilde{x}\right)$ for $\alpha =1.3$, and for various $\gamma$ values. The $T_{\alpha ,\gamma }\left(\tilde{x}\right)$ values were calculated as in Fig. 1.

Figure 3

$T_{\alpha ,\gamma }\left(\tilde{x}\right)$ for $\alpha =2$ and even $\gamma$. These functions corresponds to the Gauss-Hermite functions. The $T_{\alpha ,\gamma }\left(\tilde{x}\right)$ values were calculated as in Fig. 1.

Figure 4

$T_{\alpha ,\gamma }\left(\tilde{x}\right)$ for $\gamma =3$ and various $\alpha$ values. The $T_{\alpha ,\gamma }\left(\tilde{x}\right)$ values were calculated as in Fig. 1.

Figure 5

The strip of possible $\gamma$ values for the first correction term, $T_{\alpha ,\gamma }\left(\tilde{x}\right),$ as a function of $\alpha$ shown as the shaded area in the figure.

Figure 6

$w_S\left(\mathcal{P}\right)$ for $D=1/6$ and $n=20$ drawn in a semi-log scale. A comparison between the CLT, the exact solution, the truncated Edgeworth expansion, and the (leading order) fractional Gauss Edgeworth expansion.

Figure 7

$w_S\left(\mathcal{P}\right)$ for $D=0.3$ and $n=200$. A comparison between the CLT, the exact solution, the truncated Edgeworth expansion (coincides with the CLT for $D=0.3)$, and the (leading order) fractional Gauss Edgeworth expansion.

Figure 8

$n^{*}$ for different $D$ values in the Gaussian regime, where $n^{*}$ measures the convergence of the truncated Edgeworth expansion (blue) and the (leading order) fractional Gauss Edgeworth expansion (orange) to the exact solution. ${\varepsilon }_{\mathrm{cut}}=3\times {10}^{-4}$. The figure illustrates that the truncated Edgeworth does not work so well compared to the fractional Gauss Edgeworth expansion. In the inset we show that the leading order fractional Gauss Edgeworth indeed has a slight increment in $n^{*}$ when $D$ approaches $1/3$.

Figure 9

$n^{*}$ values for different $D$ values in the Lévy regime, where $n^{*}$ measures the convergence of the leading order fractional Edgeworth expansion to the exact solution. ${\varepsilon }_{\mathrm{cut}}=3\times {10}^{-4}$.

Figure 10

$w_S\left(\mathcal{P}\right)$ for $D=3/7$, corresponding to $\alpha =4/3$ (left-hand side) and $D=11/30$, corresponding to $\alpha =19/11$ (right-hand side) for $n=10$ (first row), $n=100$ (second row), and $n=1000$ (third row). A comparison between $L_{\alpha }\left(x\right)$, the exact solution, and the leading order fractional Lévy Edgeworth expansion.

Figure 11

$w_S\left(\mathcal{P}\right)$ for $D=11/30$ and $n=100$. A comparison between $L_{\alpha }\left(x\right)$, the exact solution, and the leading order fractional Lévy Edgeworth expansion in a log-log plot, in order to highlight the power-low decaying behavior of the tails. The black dashed line has the expected slope of $\alpha +1\simeq 3$ for $D=11/30$, and all of these curves converge to this slope for large $\mathcal{P}$.

Figure 12

$\frac{\partial log\left[w_S(\mathcal{P}=0)\right]}{\partial log\left(n\right)}$ as a function of $log\left(n\right)$ for different $D$ values. Each curve converges asymptotically to its suitable $\alpha$. The curves corresponding to $D=0.1$ and $D=0.3$ tend asymptotically to $-0.5$ as expected in the Gaussian regime, while the curve corresponding to $D=0.3$ converges for much higher $n$ values. For $D$ equals $0.4,0.5$, and $0.6$ the curves tend asymptotically to $-1/\alpha$, i.e., to $-2/3,-1$, and $-3/2$ respectively. In the Lévy regime, $D=0.5$ gives a pure Lévy density function already for $n=1$. For higher and lower $D$ values in this regime the convergence becomes slower, and higher values of $n$ are needed for the asymptotic convergence.