Hoeffding Inequality.

Question

A drunkard stands on the axis of integer numbers. He flips a coin that gets Head with p probability. If he gets Head, he goes 2 steps right, and if he get tails, he goes one step left. The drunkard flips n times, and the coin flips are independent events. Let $ X_n$ be his position after those n flips. For what value of p, $P(X_n = 0)$ can be bounded with hoeffding inequality. Calculate the bound, and compare it to the bound from using chebychev inequality.

Basically I started by naming the conditions to use hoeffding inequality, which is having independent variables and having the $Im(X) \in \{1,-1\}$ but this is not the case. So how am I suppose to do it?

Answer 1

HINT: It looks like the problem is referring to the more general version of Hoeffding's inequality: Let $X_1,\ldots,X_n$ be independent random variables with each $X_i \in [a,b]$. If we define $\bar{X} = \frac{1}{n}(X_1 + \ldots + X_n)$ then $$\mathbb{P}(|\bar{X} - E[ \bar{X}]| \geq t) \leq \exp\left(-\frac{2n^2 t^2}{n(b-a)^2} \right).$$

What does it say in your case?