Here are two different versions of Gaussian ballot theorems in the literature, each on different while similar events but the rate is quite different:


1. [P39, Probability Result 1](http://wrap.warwick.ac.uk/129549/7/WRAP-moments-functions-squareroot-cancellation-chaos-Harper-2020.pdf): For any independent sequence of Gaussian random variables $G\_m$, mean-zero and variances in between $1/20$ and $20$, and denote $S\_j:=\sum\_{m=1}^j G\_m$ we have for any $a>0$ $$\mathbb{P}(S\_j\le a,\forall j\le n)\asymp\min(1,\frac{a}{\sqrt{n}}).$$


and


2. [P19, Theorem 5](https://arxiv.org/pdf/1601.00582.pdf): For $G\_m$ as above, but now variances are the same(say $1$). Let $\delta>0$. Then there is some constant $C(\delta)>0$ such that for all $B>0$, $b\le B-\delta$ and $n\geq 1$ we have $$\mathbb{P}(S\_n\in(b,b+\delta),S\_j\le B, \forall j<n)\le C\frac{(1+B)(1+B-b)}{n^{3/2}},$$
and if $\delta<1$, $$\mathbb{P}(S\_n\in[0,\delta], S\_j\le 1,\forall 0<j<n)\geq \frac{1}{Cn^{3/2}}.$$


So I kind of don't understand, say the upper bound part, why by changing the control on last step from $\le a$ to be in some fixed range will make the rate shrink by $n$. Or the question may be phrased as, why are these two results consistent with each other?



What happens in the difference rate between these two versions of ballot theorem?
$\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$The paper linked in your post contains a reference to a reference to an apparent proof, which I hope results in an overall valid proof.


Anyhow, we can assess the plausibility of the relations  

\begin{equation\*}
 P(S\_j\le a\ \forall j\le n)\asymp\min(1,a/\sqrt{n}) \tag{1}\label{1}
\end{equation\*}
and  

\begin{equation\*}
 P(S\_n\in(b,b+\de),S\_j\le B\ \forall j<n)\le C\frac{(1+B)(1+B-b)}{n^{3/2}} \tag{2}\label{2}
\end{equation\*}
by replacing the probabilities there by the corresponding probabilities for the standard Wiener process $(W\_t)$.


Indeed, let $M\_1:=\max\_{t\in[0,1]}W\_t$. Let $\Phi$ and $\vpi$ denote the standard normal cdf and pdf, respectively. Let $\ep:=1/\sqrt n\to0$.


Then, by the reflection principle,
\begin{equation\*}
 P(M\_1\le \ep a)=2(\Phi(\ep a)-\Phi(0))\asymp\min(1,a\ep), \tag{1a}\label{1a}
\end{equation\*}
which suggests that \eqref{1} is plausible.


Now, to assess the plausibility of \eqref{2}, write  

\begin{equation\*}
\begin{aligned}
 &P(W\_1\in\ep(b,b+\de),M\_1\le \ep B) \\ 
 &=P(W\_1\in\ep(b,b+\de))-P(W\_1\in\ep(b,b+\de),M\_1>\ep B) \\ 
 &=P(W\_1\in\ep(b,b+\de))-P(W\_1\in\ep(2B-b-\de,2B-b)), 
\end{aligned}
\end{equation\*}
again by the reflection principle. So,
\begin{equation\*}
P(W\_1\in\ep(b,b+\de),M\_1\le \ep B)=g(\de)-g(0)=\de\,g'(c\de), \tag{2a}\label{2a}
\end{equation\*}
where
\begin{equation\*}
 g(t):=\Phi(\ep(b+t))-\Phi(\ep(2B-b-\de+t)) \tag{2b}\label{2b}
\end{equation\*}
and $c=O(1)$. Note that for $t=O(1)$
\begin{equation\*}
 g'(t)=\ep\,[\vpi(\ep(b+t))-\vpi(\ep(2B-b-\de+t))] \\ 
 =O(\ep^2\,|\vpi'(\ep c\_1)|)=O(\ep^2\,\ep |c\_1|\vpi(\ep c\_1))=O(\ep^3), 
\end{equation\*}
where $c\_1=O(1)$.
So,
\begin{equation\*}
P(W\_1\in\ep(b,b+\de),M\_1\le \ep B)=O(\ep^3)=O(1/n^{3/2}), \tag{2c}\label{2c}
\end{equation\*}
which suggests that \eqref{2} is plausible as well, at least as far as the order of magnitude is concerned.


Thus, looking at \eqref{1a}, we can say that $P(M\_1\le \ep a)\asymp\ep$ for small $\ep$ because $P(M\_1\le \ep a)$ is a **first-order** difference of the values of $\Phi$ for arguments near $0$ differing by $\asymp\ep$, **and $\Phi'(0)\ne0$**.


On the other hand, looking at \eqref{2a} and \eqref{2b}, we can say that $P(W\_1\in\ep(b,b+\de),M\_1\le \ep B)=O(\ep^3)$ because $P(W\_1\in\ep(b,b+\de),M\_1\le \ep B)$ is a **second-order** difference of the values of $\Phi$ for arguments differing from $0$ by $O(\ep)$, **and $\Phi''(0)=0$** (so that $\Phi''(\ep c\_2)=O(\ep)$ if $c\_2=O(1)$). The second order of the difference in question is most clearly seen in the case when $b=B-\de$, which implies that $g(t)=\Phi(\ep(B+t-\de))-\Phi(\ep(B+t))$.


