Estimating $\mathbb P\{\max_{1\le j\le n}\lvert S_j\rvert\le t\}$, so called Charles Stein's theorem? Problem
(Kai Lai Chung, A Course in Probability Theory, section 5.5, Ex6) Suppose $\{X_n\}_{n>0}$ is a sequence of i.i.d. random variables. $S_n:=X_1+\dotsb+X_n$. For each $t>0$, define $\nu(t,\omega)=\min\{n\colon\lvert S_n(\omega)\rvert>t\}$ if such an $n$ exists, or $+\infty$ if not. If $\mathbb P\{X_1\neq0\}>0$, then for every $t>0$ and $r>0$ we have $\mathbb P\{\nu(t)>n\}\le\lambda^n$ for some $\lambda<1$ and all large $n$.
It's Charles Stein's theorem, but I didn't dig out any reference online.
Thought
First, $\mathbb P\{\nu(t)>n\}=\mathbb P\{\max_{1\le j\le n}\lvert S_j\rvert\le t\}$. We need to estimate the last term. I don't know what to do next.
Kolmogorov's inequality is used to estimate such a term, but it's different: we should assume that the random variable is bounded and it's an $O(1/\sigma^2(S_n))$-bound, not $\lambda^n$.
Any idea? Thanks!
 A: I solved it, hinted by the proof on Durrett, Thm 4.4.2.
WLOG, $\mathbb P\{X_1>0\}>0$. Choose $\eta>0$ such that $\mathbb P\{X_1\ge\eta\}\ge\epsilon$ for some $\epsilon>0$. We choose $N$ such that $N\eta>2t$.
If $\max_{1\le j\le Nk}\lvert S_j\rvert\le t$, we must have $X_j+\dotsb+X_{j+N-1}\le2t<N\eta$ for $j=1,N+1,\dotsc,N(k-1)+1$. By stochastic independence, $\mathbb P\{\max_{1\le j\le Nk}\lvert S_j\rvert\le t\}\le(1-\epsilon^N)^k$. The required estimation follows.
A: Note that the set,
$$
S_1:=\bigg\{c\in\mathbb{R}:P\left(\bigcap\limits_{1\le i\le ct}\{\lvert X_i\rvert \le ct\}\right)\ge P\left(\max_{1\le j\le ct}\lvert S_j\rvert\le t\right)\bigg\}\,,
$$
is non-empty since it must contain $c=2$. Furthermore, the set,
$$
S_2:=\bigg\{s:P(\lvert X_1\rvert > st)<1\bigg\}\,,
$$
is also non-empty, since $P(\emptyset)=0\,\,$ and $\,\,P(\lvert X_1\rvert > 0)>0$.  

Consequently, for

$$\,\,\kappa{}={}\mathrm{min}\bigg\{S_1\bigcap S_2 \bigg\}\,\,\,\, \mbox{and} \,\,\,\,t>0\,,
$$


we have (for $n>\kappa t$):
$$
\begin{eqnarray*}
P\left(\nu(t)>n\right)&{}={}&P\left(\max_{1\le j\le n}\lvert S_j\rvert\le t\right){}={}P\bigg(\lvert S_1\rvert\le t,\,\ldots,\,\lvert S_n\rvert\le t\bigg) \\
&{}={}&P\bigg(\lvert X_1\rvert\le t\,,\,\lvert X_1+X_2\rvert\le t\,,\ldots,\,\lvert X_1+\ldots+X_n\rvert\le t\bigg) \\
&{}\le{}&P\bigg(\lvert X_1\rvert\le \kappa t\,,\,\lvert X_2\rvert\le \kappa t\,,\,\lvert X_3\rvert\le \kappa t\,,\ldots,\,\lvert X_n\rvert\le \kappa t\bigg)\\
&{}={}&P\bigg(\lvert X_1\rvert\le \kappa t\bigg)P\bigg(\lvert X_2\rvert\le \kappa t\bigg)\ldots P\bigg(\lvert X_n\rvert\le \kappa t\bigg)\\
&{}={}&\lambda^n\,,
\end{eqnarray*}
$$
where $$\lambda{}:={}P\bigg(\lvert X_1\rvert\le \kappa t\bigg)\,.$$
