It is a known result that an infinite one-dimensional random walk almost surely crosses every point (according to Wikipedia at least). Can this result be generalized to random walks with probabilities depending on the current position?
Formally speaking, given a function from the integers to the reals $p_n$, does a random integer sequence $a_n$, where $a_1 = 0$ and $a_{k+1}=a_k + 1$ with a probability of $p_k$ and $a_{k+1}=a_k - 1$ otherwise, almost surely hit every integer?
My intuition is that this is true for any $p_n$ where $p_k \ge 0.5$ for $k < 0$, $p_k \le 0.5 $ for $k > 0$ and $p_k \ne 0$ and $p_k \ne 1$ for any $k$. Since a normal random walk returns to $0$ infinitely many times, this random walk should too, since the probabilities are skewed towards $0$. Then since all probabilities are non-zero, there exists a non-zero probability of hitting any integer from $0$. Thus every integer is hit almost surely.
However I know that this condition isn't necessary, since the random walk defined by $p_k = 0.5$ for $k \ne 1$ and $p_k = 0.25$ for $k = 1$ also crosses any point: For any $k \le 1$ the changed probability doesn't matter (probabilites are skewed towards the negative integers), so the walk behaves just like a normal random walk, hitting every integer below $2$ infinitely many times. Eventually the sequence will cross $a_k = 2$, in fact infinitely many times since the right half also behaves like a normal random walk, eventually coming back to $a_k = 2$. From there on it follows that every integer is hit. I'm also pretty sure this logic extends to the case where you have any finite number of non-zero-or-one probabilities surrounded by infinitely many $0.5$s, or even more generally surrounded by probabilities $0.5 \le p_k \ne 1$ to the left and $0 \ne p_k \le 0.5$ to right.
Is a less strict condition for crossing every value known (or did I make a mistake in my condition)?