\le 2i-1 \le N_n-1} B_{2i-1}(w) \leq H_n (w) \leq \sum_{i=1}^{n}
B_i(w) && \mbox{ if } w_1=a, \\
\sum_{1 \le 2i \le N_n-1} B_{2i}(w) \leq H_n (w) \leq \sum_{i=1}^{n}
B_i(w) && \mbox{ if } w_1=b.\end{aligned}$$ Therefore, by large deviation results for sums of independent Bernoulli random variables (see for instance the book by Bollobas [@Bollobas Th. 7, p.13]), there exists $D_{1}>0$ such that: $$\begin{aligned}
{\mathbb P_n}\left( H_n \geq \frac94 n^\varepsilon \right) & \leq & {\mathbb P}\left(
\sum_{i=1}^{n} B_i \geq \frac94 n^\varepsilon \right) \\
& \leq & {\mathbb P}\left(\sum_{i=1}^{n} B_i - n{\mathbb E}B_1 \geq
\frac14 n^{\varepsilon} \right)
\leq O \left( \exp \left( -D_{1} n^{\varepsilon}
  \right) \right).\end{aligned}$$ In the same manner, by looking only to the $B_i$’s with odd indices (when the word begins with “$a$”) or only to the $B_i$’s with even indices (when the word begins with “$b$”) and using lemma \[Nn\] in which $\gamma=1$, we obtain the existence of $D_{2}>0$ such that: $$\begin{aligned}
&& {\mathbb P_n}\left( H_n \leq \frac14 n^\varepsilon \right) \\
& \leq & {\mathbb P}\left( N_n < \frac{n}{2} \left(1-n^{-1/2+\varepsilon}
\right)
  \right) + {\mathbb P}\left( \sum_{i=1}^{n/2\left(1-n^{-1/2+\varepsilon}
\right)} B_i\leq \frac14  n^\varepsilon
  \right)\\
& \leq & O \left( \exp \left( -C_1 n^{2\varepsilon}
  \right) \right) + O \left(  \exp \left(
    - D_{2} 