$\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$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 near $0$ differing by $\asymp\ep$, 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 $g(t)=\Phi(\ep(B+t-\de))-\Phi(\ep(B+t))$.

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

$\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$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 \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 near $0$ differing by $\asymp\ep$, 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 $g(t)=\Phi(\ep(B+t-\de))-\Phi(\ep(B+t))$.

$\newcommand{\de}{\delta}\newcommand{\vpi}{\varphi}\newcommand\ep\varepsilon$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 near $0$ differing by $\asymp\ep$, 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 $g(t)=\Phi(\ep(B+t-\de))-\Phi(\ep(B+t))$.