Reference Request: Simple Random Walk on $\mathbb Z$ is Unimodal

Question

I am looking for a reference to the following claim. Let $X = (X_t)_{t\ge0}$ be a continuous time simple random walk. Then $$ m \mapsto P(|X_t| = m) : \mathbb N \to [0,1] $$ is (weakly) decreasing (or non-increasing, if you wish). In particular, this follows from the fact that the distribution of $X_t$ is unimodal (with mode 0, and symmetric about 0).

Given the answer by Carlo, it seems that 'unimodal' only means 'has a unique mode' rather than 'has a unique global mode and no local modes (other than the global one'). To be honest, the former makes much more sense as a definition. I, however, want to show that there are no local modes either.

I can prove this, no problem, but it takes up space in my paper and the proof is not of interest to the rest of the paper; hence I'd prefer to reference the result, if at all possible. That said, I am yet to find a suitable reference, so pointers would be appreciated, thanks!

Answer 1

A proof that the simple random walk in $\mathbb{Z}^d$ is weakly unimodal is given on page 335 of this book [1].

_{[1] Khoshnevisan D., Xiao Y. (2000) Images and Level Sets of Additive Random Walks. In: Giné E., Mason D.M., Wellner J.A. (eds) High Dimensional Probability II. Progress in Probability, vol 47. Birkhäuser, Boston, MA}