What happens in the difference rate between these two versions of ballot theorem?

Question

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: 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: 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?

Answer 1

$\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$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 $g(t)=\Phi(\ep(B+t-\de))-\Phi(\ep(B+t))$.