Random Walk Upper-Bound. For any symmetric Random Walk, absolute value of partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}$?

Question

Is it possible to prove that for any symmetric Random Walk, the absolute value of its partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}\quad\large?$

I run some simulation, clearly for initial value it is not true, but for large $n$ it seems true that it is a real absolute bound.

Answer 1

No

For example you might with positive probability get nine $+1$s in a row.

But $\sqrt{18 \pi}+\sqrt{\pi / 2} \lt 9$.

More substantially, the law of the iterated logarithm says $\displaystyle \limsup_{n \to \infty} \frac{S_n}{\sqrt{2n \log\log n}} = 1$ almost surely. For big enough $n$ you will have $\dfrac{\sqrt{2n \log\log n}}{\sqrt{2n \pi} + \sqrt{\pi/2}} \gt 1$.

Answer 2

My motivation was this.

The average number of zeros for Sn is Sqrt[2/Pi] for every interval 2n+1. So if after k^2 steps the value of Sn is greater than (2k+1)*Sqrt[Pi/2] than in the next interval 2k+1 the possibility for Sn to become zero would be impossible. Something wrong?