Consider a random walk on the integers where the probability of transitioning from $n$ to $n+1$ is $p_n$ (and of course, the probability of transitioning from $n$ to $n-1$ is $1-p_n$); we assume all $p_n$ are strictly less than $1$. Note that the probability of going right is position-dependent, and we certainly do not assume that all $p_n$ are the same. Suppose we know that this random walk is pretty well concentrated; for example, let us assume that we know that for all $c \geq 0$ and for all $t$ large enough, $$P(|X(t) - (1/3)t| \geq c \sqrt{t}) \leq e^{-c^2}$$ where $X(t)$ is the state of the walk after $t$ steps.

Now suppose we increase every $p_n$ by $\epsilon$ (and correspondingly decrease the probability of transitioning from $n$ to $n-1$ by $\epsilon$), where $\epsilon$ is some number such that $p_n + \epsilon < 1$ for all $n$. Let $Y(t)$ be the state of the new chain after $t$ steps. My question is: does a similar concentration result hold for $Y(t)$?

It seems very intuitive that $Y(t)$ should concentrate around $(1/3)t + 2 \epsilon t$.

P.S. I asked this math.SE a few days ago without any answer.

moves to the righton average (like $(1/3)t$ in your example) our would a movement to the left or a stationary concentration around $0$ be permitted? (In the last cases there would probably be easy counterexamples when some $p_n$ is changed from 0 to $\epsilon$.) – Günter Rote Feb 27 '13 at 17:04