Expected distance of biased 1d random walk with 0 drift

Question

There were many question about biased 1d random walks earlier, but as far as I can tell none of these are directly related.

Let $p,q>0$ such that $p+q=1$. Let $X_0=0$ and

• $X_{i+1} = X_i+1$ with probability $p$
• $X_{i+1} = X_i-p/q$ with probability $q$

Note that $\mathbb E[X_n] = 0$ for all $n$. What is the expected distance from the origin, that is $\mathbb E[|X_n|]$? I am interested in the exact constant. One can calculate that $$c\sqrt{n}\le E\left[|X_n|\right]\leq \mathbb E\left[(X_n)^2\right]^{1/2} = \sqrt{n (p/q)}$$ for some absolute constant $c$ (depending on $p$). Is the exact constant known? Note for $p=1/2$ the constant is $\sqrt{2/\pi}$, moreover we know the full Maclaurin series. What is known for arbitrary $p$?

Many thanks!

Clarification about earlier questions:

• The question Random walk with zero drift is similar, however the direct calculation here already gives a sharper bound than the Law of the iterated logarithm.