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 we 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 
$$P(|X(t) - (1/3)t| \geq  c \sqrt{t}) \leq e^{-c^2}$$ 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][1] math.SE a few days ago without any answer.


  [1]: http://math.stackexchange.com/questions/311591/what-happens-to-a-random-walk-when-we-increase-the-probabilities-of-going-right