Expected number of maximum difference between heads and tails

Question

Let's say a fair coin is tossed $n$ times. Let $H(i)$ denote the total number of heads by the $i$-th toss and $T(i)$ denote the total number of tails by the $i$-th toss.

Define $D(i) = |H(i) - T(i)|$, the difference at $i$-th toss.

The maximum is defined as $M(n) = max^n_{i=1}D(i)$.

Now, what'd be an asymptotic bound for $E(M(n))$?

Answer 1

By Doob's L^2 inequality (see e.g. https://arxiv.org/pdf/1202.0447.pdf) we have $E(M(n)^2) \le 4E(D(n))^2)=4n $, so $E(M(n)) \le 2 \sqrt{n}$.

On the other hand, denote by $Z$ a standard normal variable. By the Central limit Theorem, $$E(M(n)) \ge E|D(n)|=\sqrt{n}E|Z|(1+o(1))=\sqrt{\frac{2n}{\pi}}(1+o(1)) \,.$$

Answer 2

$$\frac{E M(n)}{\sqrt{n}} \rightarrow \sqrt{\frac{\pi}2}\approx 1.25$$ see here