Let $X:\Omega \to \mathbb{R}$ be a random variable on a probability space $(\Omega,\mathcal{A},\mathbb{P})$ and denote by $$\chi(\xi) := \mathbb{E}e^{i \xi \cdot X}, \xi \in \mathbb{R}^d,$$ its characteristic function. I'm looking for sufficient and necessary conditions in terms of the characteristic function that (the distribution of) $X$ has a density with respect to Lebesgue measure.
For example, there are the following results:
- If $\int_{\mathbb{R}^d} |\chi(\xi)| \, d\xi < \infty$, then $X$ has a density with respect to Lebesgue measure.
- $X$ has an $L^2$-density with respect to Lebesgue measure if, and only if, $\int_{\mathbb{R}^d} |\chi(\xi)|^2 \, d\xi < \infty$.
Are there any similar statements? I'm mainly interested in the existence of the density and not in additional properties of the density (such as $L^p$-integrability, differentiability,...).
Thanks!
