Try $w_0 = b$, $w_1 = bab$, $w_2 = baba^2bab$, $w_3 = baba^2baba^3baba^2bab$, and recursively $w_{i+1} = w_ia^{i+1}w_i$.

Then I believe $\limsup_i Pr(w_i,a^n) = \frac{2^{2-n}}{3} \rightarrow_n 0$.

-Danny "Likes Anything That Resembles a Zimin Word" Rorabaugh

Try $w_0 = b$, $w_1 = bab$, $w_2 = baba^2bab$, $w_3 = baba^2baba^3baba^2bab$, and recursively $w_{i+1} = w_ia^{i+1}w_i$.

Then $\displaystyle \limsup_{i \rightarrow \infty} Pr(w_i,a^n) = \lim_{i \rightarrow \infty} \frac{2^{i + 2 - n} +n - i -3}{3\cdot 2^i - i - 2} = \frac{2^{2-n}}{3} \rightarrow_n 0$.

-Danny "Likes Anything That Resembles a Zimin Word" Rorabaugh