Is there a probability measure $\mu$ on $\mathbb{Z}$ such that, for every $0 < \alpha \leq 1$ and every finitely supported probability measure $\nu$ on $\mathbb{Z}$, it holds that the $\alpha \mu + (1-\alpha)\nu$-random walk on $\mathbb{Z}$ is recurrent? It seems like $\mu(n) = C/(1+n^2)$ is a good candidate.

Is there a probability measure $\mu$ on $\mathbb{Z}$ such that, for every $0 < \alpha \leq 1$ and every finitely supported (¹) probability measure $\nu$ on $\mathbb{Z}$, it holds that the $\alpha \mu + (1-\alpha)\nu$-random walk on $\mathbb{Z}$ is recurrent? It seems like $\mu(n) = C/(1+n^2)$ is a good candidate.

(¹) Added in edit

Is there a probability measure $\mu$ on $\mathbb{Z}$ such that, for every $0 < \alpha \leq 1$ and every probability measure $\nu$ on $\mathbb{Z}$, it holds that the $\alpha \mu + (1-\alpha)\nu$-random walk on $\mathbb{Z}$ is recurrent? It seems like $\mu(n) = C/(1+n^2)$ is a good candidate.

Is there a probability measure $\mu$ on $\mathbb{Z}$ such that, for every $0 < \alpha \leq 1$ and every finitely supported probability measure $\nu$ on $\mathbb{Z}$, it holds that the $\alpha \mu + (1-\alpha)\nu$-random walk on $\mathbb{Z}$ is recurrent? It seems like $\mu(n) = C/(1+n^2)$ is a good candidate.