It is quite challenging to ascertain whether a certain multivariable function is a characteristic function when its corresponding measure is not trivial to be or not to be a probability measure on R^d. In this article, we formulate multi-zeta function-based high-dimensional discrete probability distributions with infinitely many mass points on Z^d and Z^d-valued random walks given by those convolutions in terms of multiple zeta functions. In particular, a necessary and sufficient condition is provided for some polynomial finite Euler products to yield characteristic functions.
