Entropy Inequalities for Stable Densities and Strengthened Central Limit Theorems

Abstract

We consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function. By showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a Lévy centered density of parameter $\lambda >1$ . This introduces a new information-theoretic approach to the central limit theorem for stable laws, in which the main argument is shown to be the relative fractional Fisher information, recently introduced in Toscani (Ricerche Mat 65(1):71–91, 2016). In particular, it is proven that, with respect to the relative fractional Fisher information, the Lévy density satisfies an analogous of the logarithmic Sobolev inequality, which allows to pass from the monotonicity and decay to zero of the relative fractional Fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate.

Appendix

In this short appendix we will collect the notations concerning fractional derivatives, with some applications to the Lévy stable laws. Also, we list the various functional spaces used in the paper. Last, we recall some properties of Linnik distribution, that represents the main example of probability density to which the results of the present paper can be applied.

1.1 Appendix 1: Fractional Derivatives

For $0<\nu < 1$ we let $R_\nu$ be the one-dimensional normalized Riesz potential operator defined for locally integrable functions by [42, 47]

$$\begin{aligned} R_\nu (f)(x) = S(\nu ) \int _{\mathbb {R}}\frac{f(y)\, dy}{|x-y|^{1-\nu }}. \end{aligned}$$

The constant $S(\nu )$ is chosen to have

$$\begin{aligned} \widehat{R_\nu (f)}(\xi ) = |\xi |^\nu \widehat{f}(\xi ). \end{aligned}$$

(57)

Since for $0<\nu < 1$ it holds [33]

$$\begin{aligned} \mathcal F |x|^{\nu -1} = |\xi |^{-\nu } \pi ^{1/2} \Gamma \left( \frac{1-\nu }{2} \right) \Gamma \left( \frac{\nu }{2} \right) , \end{aligned}$$

(58)

where, as usual $\Gamma (\cdot )$ denotes the Gamma function, the value of $S(\nu )$ is given by

$$\begin{aligned} S(\nu ) = \left[ \pi ^{1/2} \Gamma \left( \frac{1-\nu }{2} \right) \Gamma \left( \frac{\nu }{2} \right) \right] ^{-1}. \end{aligned}$$

Note that $S(\nu ) = S(1-\nu )$.

We then define the fractional derivative of order $\nu$ of a real function f as ($0<\nu < 1$)

$$\begin{aligned} \frac{d^\nu f(x)}{dx^\nu } = {\mathcal {D}}_\nu f(x) = \frac{d}{dx}R_{1-\nu }(f)(x). \end{aligned}$$

(59)

Thanks to (57), in Fourier variables

$$\begin{aligned} \widehat{{ {\mathcal {D}}}_\nu f}(\xi ) = i \frac{\xi }{|\xi |} |\xi |^\nu \widehat{f}(\xi ). \end{aligned}$$

(60)

It is immediate to verify that, for $0<\nu <1$, Lévy centered stable laws $\omega (x)$ of parameter $\lambda = 1+\nu$ satisfy

$$\begin{aligned} \frac{ {\mathcal {D}}_\nu \, \omega (x)}{\omega (x)} = -\frac{x}{1+\nu }. \end{aligned}$$

(61)

Indeed, identity (61) is verified if and only if, on the set $\{f>0\}$

$$\begin{aligned} {\mathcal {D}}_\nu f(x) = -\frac{xf(x)}{1+\nu }. \end{aligned}$$

(62)

Passing to Fourier transform, this identity yields

$$\begin{aligned} i \xi |\xi |^{\nu -1} \widehat{f}(\xi ) = -i \frac{1}{1+\nu }\frac{\partial \widehat{f}(\xi )}{\partial \xi }, \end{aligned}$$

and from this follows

$$\begin{aligned} \widehat{f}(\xi ) = \widehat{f}(0) \exp \left\{ -|\xi |^{\nu +1}\right\} . \end{aligned}$$

(63)

Finally, imposing that f(x) is a probability density function (i.e. by fixing $\widehat{f}(\xi = 0) =1$), we obtain that the Lévy stable law of order $1 + \nu$ is the unique probability density solving (61)

1.2 Appendix 2: Functional Framework

We list below the various functional spaces used in the paper. For $p \in [1, +\infty )$ and $q \in [1, +\infty )$, we denote by $L_q^p$ the weighted Lebesgue spaces

$$\begin{aligned} L^p_q := \left\{ f : {\mathbb {R}}\rightarrow {\mathbb {R}}\text { measurable; }\Vert f\Vert _{L^p_q}^p := \int _{\mathbb {R}}|f(x)|^p \, (1+ x^2)^{q/2} \, dx < \infty \right\} . \end{aligned}$$

In particular, the usual Lebesgue spaces are given by

$$\begin{aligned} L^p := L^p_0.\end{aligned}$$

Moreover, for $f \in L^1_q$, we can define for any $\kappa \le q$ the $\kappa ^{th}$ order moment of f as the quantity

$$\begin{aligned} m_\kappa (f):= \int _{\mathbb {R}}f(x) \, |x|^{\kappa } dx\, < \, \infty . \end{aligned}$$

For $k \in \mathbb {N}$, we denote by $W^{k,p}$ the Sobolev spaces

$$\begin{aligned} W^{k,p} := \left\{ f \in L^k; \Vert f\Vert _{W{k,p}}^p := \sum _{|j| \le k} \int _{{\mathbb {R}}} \left| f^{(j)}(v)\right| ^p \, dx < \infty \right\} . \end{aligned}$$

If $p=2$ we set $H^k := W^{k,2}$.

Given a probability density f, we define its Fourier transform $\mathcal F(f)$ by

$$\begin{aligned} \mathcal F(f)(\xi ) = \widehat{f}(\xi ) := \int _{\mathbb {R}}e^{- i \, \xi \, x} f(x)\, dx, \qquad \forall \xi \in {\mathbb {R}}. \end{aligned}$$

The Sobolev space $H^k$ can equivalently be defined for any $k \ge 0$ by the norm

$$\begin{aligned} \Vert f\Vert _{H^{k}} := \left\| \mathcal F\left( f \, \right) \right\| _{L^2_{2k}}. \end{aligned}$$

The homogeneous Sobolev space $\dot{H}^k$ is then defined by the homogeneous norm

$$\begin{aligned} \Vert f\Vert _{\dot{H}^k}^2 := \int _{\mathbb {R}}|\xi |^{2 k} \left| \widehat{f}(\xi ) \right| ^2 \, d \xi . \end{aligned}$$