[Paper Review] A Cramér--Wold device for infinite divisibility of $\mathbb{Z}^d$-valued distributions

David Berger, Alexander Lindner|arXiv (Cornell University)|Nov 17, 2020

Probability and Risk Models16 references4 citations

TL;DR

This paper establishes a Cramér–Wold device for infinite divisibility of $\mathbb{Z}^d$-valued distributions, proving that such a distribution is infinitely divisible if and only if all its linear projections $a^T X$ are infinitely divisible for all $a \in \mathbb{R}^d$, or equivalently, for all $a \in \mathbb{N}_0^d$. The key technical tool is a Lévy–Khintchine representation with a signed Lévy measure for zero-free characteristic functions of $\mathbb{Z}^d$-valued distributions.

ABSTRACT

We show that a Cramér--Wold device holds for infinite divisibility of $\mathbb{Z}^d$-valued distributions, i.e. that the distribution of a $\mathbb{Z}^d$-valued random vector $X$ is infinitely divisible if and only if $\mathcal{L}(a^T X)$ is infinitely divisible for all $a\in \mathbb{R}^d$, and that this in turn is equivalent to infinite divisibility of $\mathcal{L}(a^T X)$ for all $a\in \mathbb{N}_0^d$. A key tool for proving this is a Lévy--Khintchine type representation with a signed Lévy measure for the characteristic function of a $\mathbb{Z}^d$-valued distribution, provided the characteristic function is zero-free.

Motivation & Objective

• To establish a Cramér–Wold device for infinite divisibility in the context of $\mathbb{Z}^d$-valued distributions.
• To resolve the absence of a general Cramér–Wold device for infinite divisibility in multivariate settings by restricting to integer-valued distributions.
• To provide a characterization of infinite divisibility via linear projections with coefficients in $\mathbb{R}^d$ and $\mathbb{N}_0^d$, ensuring the projections remain integer-valued.
• To develop a Lévy–Khintchine-type representation with a signed Lévy measure for zero-free characteristic functions of $\mathbb{Z}^d$-valued distributions.
• To show that the signed Lévy measure in this representation must be non-negative (i.e., a true Lévy measure) if all one-dimensional projections are infinitely divisible, thereby implying infinite divisibility of the full distribution.

Proposed method

• Derive a Lévy–Khintchine-type representation for the characteristic function of a $\mathbb{Z}^d$-valued distribution using a signed finite measure supported on $\mathbb{Z}^d \setminus \{0\}$, under the assumption that the characteristic function is zero-free.
• Use the fact that the characteristic function is $2\pi$-periodic in each coordinate to extend zero-freeness from a local region to the entire $\mathbb{R}^d$.
• Apply the known uniqueness of the signed Lévy measure in the Lévy–Khintchine representation to show that if all one-dimensional projections are infinitely divisible, then the signed measure must be non-negative.
• Leverage the linear independence over $\mathbb{Q}$ of the components of a direction vector $a \in \mathbb{R}^d$ to ensure that the pushforward measure $\nu_a$ captures the full support structure of the original signed measure $\nu$.
• Use weak convergence of distributions and the closedness of the class of infinitely divisible distributions under weak convergence to extend infinite divisibility from a dense set of directions to all directions in $\mathbb{R}^d$.
• Apply the structure of lattice distributions and the invariance of infinite divisibility under affine transformations to extend the result to distributions on shifted lattices $v + A\mathbb{Z}^d$.

Experimental results

Research questions

• RQ1 Does a Cramér–Wold device hold for infinite divisibility of $\mathbb{Z}^d$-valued distributions, i.e., is the full distribution infinitely divisible if all linear projections $a^T X$ are infinitely divisible for all $a \in \mathbb{R}^d$?
• RQ2 Can the Cramér–Wold condition be restricted to $a \in \mathbb{N}_0^d$ without leaving the class of integer-valued distributions, and does this still characterize infinite divisibility?
• RQ3 Under what conditions does a Lévy–Khintchine representation with a signed Lévy measure for the characteristic function of a $\mathbb{Z}^d$-valued distribution imply that the measure is actually a non-negative Lévy measure?
• RQ4 Can the zero-freeness of the characteristic function be established via the infinite divisibility of projections in a dense set of directions, and how does this relate to periodicity and global zero-freeness?
• RQ5 Does the infinite divisibility of all projections $a^T X$ for $a \in \mathbb{N}_0^d$ imply the infinite divisibility of the full $\mathbb{Z}^d$-valued distribution?

Key findings

• A $\mathbb{Z}^d$-valued distribution is infinitely divisible if and only if all its linear projections $a^T X$ are infinitely divisible for all $a \in \mathbb{R}^d$, establishing a full Cramér–Wold device for this class.
• The condition can be restricted to $a \in \mathbb{N}_0^d$, ensuring that the projections remain $\mathbb{N}_0$-valued, and the infinite divisibility of the full distribution is still equivalent to the infinite divisibility of these projections.
• A Lévy–Khintchine-type representation with a signed Lévy measure exists for the characteristic function of a $\mathbb{Z}^d$-valued distribution if the characteristic function is zero-free.
• If all one-dimensional projections $a^T X$ are infinitely divisible and the characteristic function is zero-free, then the signed Lévy measure must be non-negative, implying the original distribution is infinitely divisible.
• The result extends to distributions on affine lattices $v + A\mathbb{Z}^d$ via affine invariance, preserving the Cramér–Wold equivalence.
• The proof relies on the closedness of the class of infinitely divisible distributions under weak convergence and the periodicity of characteristic functions on $\mathbb{Z}^d$-lattices to propagate zero-freeness globally.