I toss a fair coin $n$ times. Some notation:
$S_i=$ difference between #heads and #number of tails after the first $i$ tosses, $1\leq i\leq n$.
$M_n=\max(S_1,S_2,\dots,S_n)$,
$m_n=\min(S_1,S_2,\dots,S_n)$.
$M=\max(M_n,|m_n|)$.
Now, suppose that I know that exactly half the times heads came up. That is, $S_n=0$.
I want to say something about $M$. I want to know if it's true that $P(M<c\sqrt{n}\mid S_n=0)=\Omega_c(1)$, for an absolute constant $c>0$?
That is, do I have some positive probability, not depending on $n$, that the difference between the heads and the tails in absolute value at no stage exceeded, say, $\frac{1}{1000}\sqrt{n}$?
I know it's true if I look just at one-sided difference, that is, not in absolute value. pages 18-19 here, for example.
http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf
But not my case.