Weak convergence of complex-valued measure for bi-product path space induced by quantum walk

In this paper, a complex-valued measure of bi-product path space induced by quantum walk is presented. In particular, we consider three types of conditional return paths in a power set of the bi-product path space (1) $\Lambda \times \Lambda$, (2) $\Lambda \times \Lambda'$ and (3) $\Lambda'\times \Lambda'$, where $\Lambda$ is the set of all $2n$-length ($n\in \mathbb{N}$) return paths and $\Lambda'(\subseteq \Lambda)$ is the set of all $2n$-length return paths going through $nx$ ($x\in [-1,1]$) at time $n$. We obtain asymptotic behaviors of the complex-valued measures for the situations (1)-(3) which imply two kinds of weak convergence theorems (Theorems 1 and 2). One of them suggests a weak limit of weak values.