# Bounding the number of visits for each site of a random walk by a sequence

Recently, I asked if, for each $k>1$, a transient random walk visits each site less than $k$ times a.s.. You can find the question here: Visits from a transient random walker on the integers

This claim is true, but it got me wondering about a similar problem. Does there exist an integer sequence $\{n_k\}^\infty_{k=1}$ such that $$P[V_k\leq n_k\text{ for all }k\in\mathbb{N}]>0,$$ where, for each $k \in \mathbb{Z}$, $V_k$ is the number of visits to site $k$ from a right-transient ($p>\frac12$) RW on the integers?

The sequence $(V_k)_{k\geqslant1}$ is stationary and $P[V_1=\infty]=0$ hence for every $k$ there exists $m_k$ such that $P[V_1\gt m_k]\leqslant1/k^2$. The series $\sum\limits_kP[V_k\gt m_k]=\sum\limits_kP[V_1\gt m_k]$ converges hence, by Borel-Cantelli, $V_k\leqslant m_k$ almost surely for every $k$ except finitely many. Thus, $$P[\forall k\gt K,V_k\leqslant m_k]\to1,$$ when $K\to\infty$. Again by stationarity, for every $K$, $$P[\forall k\geqslant 1,V_k\leqslant m_{k+K}]=P[\forall k\gt K,V_k\leqslant m_k].$$ Thus, for every $p\lt1$, choosing $(n_k)=(m_{k+K})$ for $K$ large enough ensures that $$P[\forall k\geqslant 1,V_k\leqslant n_{k}]\geqslant p.$$