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   \title[Probabilities to stay between  boundaries]
    {\large On Accuracy of Approximations 
    \\ for Wiener Probabilities to stay 
    \\ between  square root boundaries}
    
 %   О точности аппроксимаций винеровских вероятностей находиться между границами порядка квадратного корня
 

\author[A.I. SAKHANENKO]{{\bf A.I. Sakhanenko}}%\protect\orcidicon{0000-0001-8827-2630}}



\address{Alexander  Ivanovich Sakhanenko  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{\textcolor{blue}{aisakh@mail.ru}}%


\thanks{\sc Sakhanenko, A.I.,
On Accuracy of Approximations 
     for Wiener Probabilities to stay 
    between  square root boundaries}
\thanks{\copyright \ 2025 Sakhanenko A.I.}
\thanks{\rm Работа выполнена при поддержке программы фундаментальных научных
 исследований СО РАН, проект FWNF-2022-0010}
\thanks{\it Received October, 14, 2025, Published December, 31, 2025}%


\semrtop \vspace{1cm}
\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc   N.S. Arkashov}
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} 
Several asymptotic formulas for
probabilities of the Wiener process  to stay 
    between different square root boundaries
    are known after papers of
    Breiman (1965), Sato (1977), Novikov (1979, 1981), G\"artner (1982),  Uchiyama (1980), Greenwood and Perkins (1983).    
   In the present work we investigate  accuracy of these  approximations.
   We present several general estimates in the
case of two boundaries. In particular, these estimates contain the
ones obtained earlier by Uchiyama in the limiting case of one boundary.
 
 
%   We  improve and generalize the estimate obtained earlier by Uchiyama 
%   in the case of one boundary. 
\medskip

%Несколько асимптотических формул для
%вероятностей пребывания винеровского процесса
%между различными границами порядка квадратного корня
%известны после работ
%Бреймана (1965), Сато (1977), Новикова (1979, 1981), Гёртнера (1982), Учиямы (1980), Гринвуда и Перкинса (1983).
%В настоящей работе мы исследуем точность этих приближений.
%Мы улучшаем и обобщаем оценку, полученную ранее Учиямой
%в случае одной границы.


\noindent{\bf Keywords:} 
 Brownian motion, Wiener process, square root boundaries, exit time, accuracy of approximation  
 \end{quote}
 
% Броуновское движение, винеровский процесс, границы порядка квадратного корня, время выхода, точность приближения
}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and main results}

\subsection{Introduction}
For a standard  Brownian motion $B_t=B(t)$,
 defined for all $t\ge0$,
introduce into consideration the following 
stopping time:
\begin{align*}           %                                    \label{i1}
\tau_{c_1,c_2}:=\inf\{t\ge1: B_t\notin (c_1\sqrt{t},c_2\sqrt{t})\}
=\inf\{t\ge1: U_t\notin (c_1,c_2)\},
\end{align*}
where $U_t=U(t):=B(t)/\sqrt t$ for $t>0$ and $c_1<c_2$.
Our main aims  are to approximate, for large $T>1$, the probability
\begin{align*}                       %                            \label{i2}
\pr ( \tau_{c_1,c_2}>T | B_1 = a)&=\pr ( \tau_{c_1,c_2}>T | U_1 = a)
\\&=
 \pr ( c_1 <U_t < c_2\  \forall t\in [1,T] | U_1 = a) 
\end{align*}
and the more general probability
\begin{align}                                                   \label{i3}
P_{c_1,c_2}(a,T,y) = \pr (\tau_{c_1,c_2}>T \quad \mbox{and} \ \ U_T\le y | U_1 = a),\quad T>1,
\end{align}
for all  parameters $a,c_1,c_2,y$ satisfying the following relations:
\begin{gather}                                   \label{i5}
  -\infty<c_1<c_2\le\infty,\quad a,y<\infty \quad \mbox{and} \quad  c_1\le a,y\le c_2.
\end{gather} 
Denote by $\D$ the set of all values $(a,c_1,c_2,y)$ which satisfy conditions from~\eqref{i5}.
We fix the numbers $c_1$ and $c_2$ and we often omit the dependence of our notations on $c_1,c_2$.
We everywhere suppose that $T > 1$.

The study of first-passage times over one of the two square root boundaries for the Brownian motion was initiated by Breiman~\cite{Breiman65}. 
After that several results in exit problems with one-sided boundaries were obtained by Sato~\cite{Sato77}, Novikov \cite{Novikov79, Novikov81}, G\"artner~\cite{Gaertner82}, Uchiyama~\cite{U80}.  
For more detailed history see  also~\cite{DHSW22}.
 All these results were summarized by Greenwood and Perkins \cite{GP83} in their Lemma 3. We recall the corresponding assertions below in our Lemmas~\ref{L1} and~\ref{L5}.
  In particular, for all $c_1<c_2$
   there exist functions
\begin{align}                                                   \label{i6}
\lambda(c_1,c_2)>0\qquad \mbox{and} \quad \psi_{c_1,c_2}(x)>0 \quad\forall x\in(c_1,c_2) 
\end{align}
 such that
 \begin{align}                                                   \label{i7}
T^{\lambda(c_1,c_2)}P_{c_1,c_2}(a,T,y) \to\Psi_{c_1,c_2}(a,y)\:=\psi_{c_1,c_2}(a)\theta_{c_1,c_2}(y)\quad\text{on}\ \ \D
\end{align}
as $T\to\infty$, where for all  $y\in(c_1,c_2)$ 
\begin{align}                                                   \label{i8}
\theta_{c_1,c_2}(c_1)=0<\theta_{c_1,c_2}(y)
:=\int_{c_1}^{y} \frac{\psi_{c_1,c_2}(x)}{||\psi_{c_1,c_2}||^2}e^{-x^2/2}dx<\theta_{c_1,c_2}(c_2) <\infty
%\ \ \forall y\in(c_1,c_2) 
\end{align}
and \ \ $0<||\psi_{c_1,c_2}||^2:=\int_{c_1}^{c_2} \psi^2_{c_1,c_2}(x)e^{-x^2/2}dx<\infty$.

Unfortunately, the explicit forms of functions 
$\lambda(c_1,c_2)$ and $\psi_{c_1,c_2}(x)$
are known only in exceptional cases. For example, Breiman~\cite{Breiman65} pointed out that
$$
\lambda(-1,1)=1\ \quad \mbox{and} \quad \lambda(-c,c)=2
\ \quad \mbox{for} \quad c^2=3-\sqrt 6. 
$$
Several authors (see, for example, ~\cite{DSW16}) also noted that 
\begin{gather*}
\lambda(0,\infty)=1/2\qquad \mbox{and} \qquad \psi_{0,\infty}(x)=\sqrt{2/\pi}x
\\
\quad \mbox{with} \qquad \theta_{0,\infty}(x)=1-e^{-x^2/2} \qquad \mbox{for} \quad x\ge0,
\end{gather*}
where $||\psi_{0,\infty}||=\sqrt[4]{2/\pi}$.


 \subsection{Statement of the problem}
 There  arises a natural task  to obtain a rate of convergence in \eqref{i7}, i.e. to find estimates for the difference
\begin{align}                                                   \label{i11}
\Delta_{c_1,c_2}(a,T,y):=T^{\lambda(c_1,c_2)}P_{c_1,c_2}(a,T,y) -\Psi_{c_1,c_2}(a,y) 
\quad\text{on}\ \ \D,
\end{align}
when $T>1$. In the particular case, when $c_2=y=\infty$, such estimate was obtained 
 %The next important step in the study of the stopping times $\tau_g(u)$ was done 
 by Uchiyama~\cite{U80}.
In his  Theorem 1.1 for all $c:=c_1\in\R$ he found a remarkable inequality:
\begin{align}                                              \label{i12}
\big|\Delta_{c,\infty}(a,T,\infty)\big|&=\big|T^{\lambda(c,\infty)}\pr ( \tau_{c,\infty}>T | U_1 = a)-
\psi_{c,\infty}(a)\theta_{c,\infty}(\infty)\big|
\\ \nonumber
&<\frac{C_1(\ee,c)}{T^{\kk(c)}}\exp(\ee x^2)
\quad\text{for any}\ \ \ee>0 \ \ \text{and} \ \ T>T_1(\ee,c)>1, % \quad\text{and all}
%\ \ c\in\R,
\end{align}
where the constants $C_1(\ee,c)$ and $T_1(\ee,c)$ depend only on the chosen numbers $\ee>0$ and $c\in\R$ (for the implicit definitions of these constants see formula (1.10) in \cite{U80}); in addition
\begin{align}                                              \label{i13}
\text{for all $c\in\R$ the function $\kk(c)$ is continuous and \ $\kk(c)>1/2$.}
\end{align}

Our aim is to improve and generalize the estimate \eqref{i12} of Uchiyama. 
 
 \subsection{Main estimates}
Below we consider only parameters from \eqref{i5}.
\begin{theorem}                                      \label{T-1}
There exists a number $\kk(c_1,c_2)>0$ such that for any  function $u_a>1$ we have on $\D$ that
\begin{align}                                                \label{i15}
\big|\Delta_{c_1,c_2}(a,T,y)\big| \le \sqrt[4]{e(1+8\Lambda(c_1,c_2))}\frac{u_a^{\Lambda(c_1,c_2)}}{ T^{\kk(c_1,c_2)}} 
\sqrt[4]{\frac{u_a^2}{u_a^2-1}}\exp\left( \frac{a^2}{2(u_a +1)} \right)
%\qquad\text{on}\ \ \D,
\end{align}
for all $T>1$, where $\Lambda(c_1,c_2):=\lambda(c_1,c_2)+\kk(c_1,c_2)$. 
\end{theorem}

Now consider several particular cases. 
\begin{corollary}                                      \label{C-1}
Let $c_2<\infty$. Then from \eqref{i15}  with $u_a=2$ 
 we immediately have 
\begin{align}                                             \label{i16}
\forall\,T>1\quad\big|\Delta_{c_1,c_2}(a,T,y)\big| \le 
\frac{C_2(c_1,c_2)}{ T^{\kk(c_1,c_2)}}
\quad\text{on}\ \ \D,
\end{align}          
where
 %(with $c_1^2\vee c_2^2=\max\{c_1^2, c_2^2\}$)
\begin{align*}    %                                            \label{i17}
C_2(c_1,c_2):= \sqrt[4]{4e(1+8\Lambda(c_1,c_2))}2^{\Lambda(c_1,c_2)}
\exp\left( \frac{\max\{c_1^2, c_2^2\}}6 \right)<\infty.
\end{align*}
\end{corollary}

Unfortunately, we cannot find the optimal function $u_a > 1$ which
minimize the right-hand side in \eqref{i15}.
Instead, we present several simple 
 approximations to such functions.
It is clear, that estimate \eqref{i16} is bad for large values of $|c_1|$, $|c_2|$, or $|a|$; for example, $C_2(c_1,\infty)=\infty$. To avoid this problem, instead of the simple $u_a=2$ we should use a function $u_a$ which depend on $a$ and $\Lambda(c_1,c_2)$, especially for large values of $|a|$ and $\Lambda(c_1,c_2)$.
Such appropriate functions $u_a$ will be found in subsection 2.7;
 as a result, we obtain the following sufficiently sharp and general estimate.

\begin{corollary}                                      \label{C-2}
For any numbers $b_a>0$ and $T>1$
\begin{align}                                               \label{i18}
\big|\Delta_{c_1,c_2}(a,T,y)\big|\le\sqrt{e^{1+1/b_a}(1+8\Lambda(c_1,c_2))}
\frac{ (1+b_aa^2)^{\Lambda(c_1,c_2)}}{ T^{\kk(c_1,c_2)}}
\quad\text{on} \ \D.
\end{align}
\end{corollary}

In particular, in the simplest case, when $b_a=1$,
\begin{align}                                               \label{i19}
\forall\,T>1\quad\big|\Delta_{c_1,c_2}(a,T,y)\big|\le e\sqrt{1+8\Lambda(c_1,c_2)}
\frac{ (1+a^2)^{\Lambda(c_1,c_2)}}{ T^{\kk(c_1,c_2)}}
\quad\text{on}\ \ \D.
\end{align}
On the other hand, with $b_a=1+\ee /\Lambda(c_1,c_2)$ we find from \eqref{i18} the next two interesting estimates.
\begin{corollary}                                      \label{C-3}
For any $\ee>0$ and $T>1$ 
\begin{align}                                               \label{i20}
\big|\Delta_{c_1,c_2}(a,T,y)\big|&\le
\frac{ C_3(\ee,\Lambda(c_1,c_2))}{ T^{\kk(c_1,c_2)}}
\left(1+\frac{\ee a^2}{\Lambda(c_1,c_2)} \right)^{\Lambda(c_1,c_2)}
\\                                              \label{i21}
&\le\frac{ C_3(\ee,\Lambda(c_1,c_2))}{ T^{\kk(c_1,c_2)}}\exp\left(\ee a^2 \right)
\qquad\text{on}\ \ \D,
\end{align}
where
\begin{align}                                               \label{i22}
C_3(\ee,\Lambda)  := \sqrt{e(1+8\Lambda)}
\exp\left( \frac{\Lambda}{2\ee} \right)<\infty.
\end{align}
\end{corollary}

Note that if $|a|\ge2$ and $\Lambda(c_1,c_2)\ge e$ then
estimate \eqref{i20} with $\ee=1/2$ is sharper than the simple inequality  \eqref{i19}.
Observe also that in \eqref{i18}, \eqref{i19}, and~\eqref{i22} we have the coefficient $ \sqrt{1+8\Lambda} $ instead of $\sqrt[4]{1+8\Lambda} $ in the other places of the paper (see the proof of Corollary~\ref{C-2} for the reason).


\begin{remark}                                      \label{R-1}
If $c_2=\infty$ then $\Lambda(c_1,\infty)<\infty$
and $\kk(c_1,\infty)=\kk(c_1)$, where  the function $\kk(\cdot)$ satisfies conditions \eqref{i13}. 
In this case inequalities  \eqref{i19} and \eqref{i20} are sharper  than estimate \eqref{i12} of Uchiyama, at least for sufficiently large $|a|$.
In addition, 
rough inequality~\eqref{i21}
is more convenient than \eqref{i12} because it  takes place for all $T>1$ with explicit constant $C_3$.
 
%In addition, o
Our considerations show that sharper estimates \eqref{i19} and \eqref{i20}
 would allow to simplify and shorten several proofs in   \cite{DHSW22} and   \cite{GP83}.
\end{remark} 

  \subsection{Key estimates}
  Introduce a  function:
  \begin{align}                                                                 \label{i23}
\ff_{v}(y) :=  \sqrt[4]{\frac{v^2}{v^2-1}}\exp\left( \frac{y^2}{2(v+1)} \right)\hspace{-1mm},
\quad v>1,\quad y\in\R.
\end{align}
  
\begin{theorem}                                      \label{T-Z}
Let a real number $T$ and functions $u_a$ and $v_y$ be such that
\begin{align}                                                        \label{i24}
  u_a>1, \quad  v_y>1 \quad \mbox{and} \quad T\ge u_av_y
  \quad\text{on}\ \ \D.
\end{align}
Then we have the following estimate:
\begin{align}                                                \label{i25}
\big|\Delta_{c_1,c_2}(a,T,y)\big| \le \frac{u_a^{\Lambda(c_1,c_2)}\ff_{u_a}(a)}{\sqrt{2\pi} T^{\kk(c_1,c_2)}} 
\int_{c_1}^y v_x^{\Lambda(c_1,c_2)}\ff_{v_x}(x) e^{-x^2/2}dx
\quad\text{on}\ \D,
\end{align}
where the numbers  $\Lambda(c_1,c_2)>\kk(c_1,c_2)>0$ were introduced in  Theorem~\ref{T-1}.
\end{theorem}

The following rough inequalities may be useful in the case of small values of $T$.
\begin{theorem}                                      \label{TT-Z}
With the  numbers $\lambda(c_1,c_2)$ defined in~\eqref{i7}
\begin{align}                                                \label{i26}
T^{\lambda(c_1,c_2)}\ge\Delta_{c_1,c_2}(a,T,y)\ge 
-\Psi_{c_1,c_2}(a,y)
 \qquad\text{on}\ \ \D.
\end{align}

In addition, for all  functions $u_a$ and $v_y$ such that
\begin{align}                                                        \label{i27}
  u_a>1 \quad \mbox{and} \quad  v_y>1
  \quad\text{on}\ \ \D
\end{align}
we have the next estimate:
\begin{align}                                                \label{i28}
0\le\Psi_{c_1,c_2}(a,y) \le \frac{u_a^{\lambda(c_1,c_2)}\ff_{u_a}(a)}{\sqrt{2\pi} }
\int_{c_1}^y v_x^{\lambda(c_1,c_2)}\ff_{v_x}(x) e^{-x^2/2}dx
\quad\text{on}\  \D.
\end{align}
\end{theorem}

\begin{remark}                                      \label{R-Z}
The limiting function $\Psi_{c_1,c_2}$,
introduced in  \eqref{i7},
always exists and is uniquely defined on $\D$.
But the function $\psi_{c_1,c_2}$,
used in all our results, may be defined only
 up to a positive constant multiplier.
 For example, we may put
 \begin{align}                                                \label{i30}
 \psi_{c_1,c_2}(a)=C_{c_1,c_2}\Psi_{c_1,c_2}(a,c_2)\quad\text{on}\ \ \D
\end{align} 
 with an arbitrary constant $C_{c_1,c_2}$ dependent only on $c_1$ and $c_2$.
 
 These facts will be established  in Lemma \ref{L5} below.
\end{remark} 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proofs }
\subsection{Main representations} Introduce into consideration an Ornstein-Uhlenbeck process
\begin{gather}                                                \label{j1}
    \omega_t := U(e^t)=e^{-t/2}B(e^{t})\quad  \mbox{for} \quad t\ge0, \quad \mbox{and let} \quad s := \log T >0.
\end{gather}
 It is well-known that there exists a function $ q_{c_1,c_2}(\cdot,\cdot,\cdot) $ such that in the domain $\D$ for all measurable $A\subset[c_1,c_2]$ and $\mu(y):=e^{-y^2/2}$
\begin{align}                                                   \label{j2}
\pr (c_1<\omega_t < c_2 \ \forall t \in [0,s] \ \ \mbox{and} \ \omega_s \in A | \omega_0 = a)=\int_A q_{c_1,c_2}(a,e^s,y)\mu(y) dy.
\end{align}
This is the probability that the Ornstein-Uhlenbeck process with the absor\-bing barriers at
positions $c_1<c_2$ is not absorbed until the time $s =\log T$.
Here the function $ q_{c_1,c_2}(\cdot,\cdot,\cdot) $ has several remarkable properties which were used, for example, in \cite{Breiman65} and \cite{U80}. 
The next assertion follows from more general Pro\-po\-si\-tion~2 and Lemma 3 in \cite{GP83}.
See also \cite{U80} for more details when $c_2=\infty$.

\begin{lemma}                                      \label{L1}
Let the real numbers $a,c_1,c_2,y$ satisfy  conditions \eqref{i5}. 
Then the function  $ q_{c_1,c_2}(\cdot,\cdot,\cdot)$, defined in  \eqref{j2}, has the following properties:

{\rm\bf (A)} for each fixed $T=e^s>1$ we have the next representation:
\begin{gather}                                                \label{j3}
    q_{c_1,c_2}(a,T,y) = \sum\limits_{k=0}^{\infty} e^{-\lambda_k s}\phi_k(a) \phi_k(y)
= \sum\limits_{k=0}^{\infty} \frac{\phi_k(a) \phi_k(y)}{T^{\lambda_k}}
\end{gather}
for some functions $ \{\phi_k(\cdot)=\phi_k(\cdot,c_1, c_2)\} $ and  numbers \mbox{$\{\lambda_k=\lambda_k(c_1, c_2) >0\}$} generated by the Ornstein-Uhlenbeck process, 
where the series converges ab\-so\-lu\-te\-ly, uniformly for $(a, y)$ in compact subsets of $[c_1, c_2]^2$.
The convergence also holds in the space % $L^2_{c_1,c_2}$ 
of all square-summable (with respect to the measure $\mu(a)\mu(y)dady$)
 functions in the plane domain $c_1<a,y<c_2$.

{\rm\bf (B)} The function $\lambda_0(c_1,c_2)$ is strictly positive, jointly continuous on the set
$$C:= \{(c_1, c_2):-\infty<c_1<c_2\le \infty\};
$$
it is also strictly increasing in $c_l\in (-\infty, c_2]$, and strictly decreasing in \mbox{$c_2\in[c_1,\infty]$}. In addition,
$$\lim_{(c_1,c_2)\to(-\infty,\infty) } \lambda_0(c_1,c_2)=0
\quad \mbox{and} \quad  
\lim_{(c_1,c_2)\to(0,0) }\lambda_0(c_1,c_2) =\infty.
$$

{\rm\bf (C)} For each $k>0$ and all $(c_1,c_2)\in C$
$$0<\lambda_0(c_1,c_2)<\dots<\lambda_k(c_1,c_2)<\lambda_{k+1}(c_1,c_2)<\infty.
$$
Moreover, all functions $\lambda_k(c_1,\infty)$ are continuous on $\R$ with
$$\lambda_{k+1}(c_1,\infty)-\lambda_k(c_1,\infty)>1/2
\quad \mbox{for all} \quad k=0,1,\dots \quad \mbox{and} \ \ c_1\in\R. 
$$

{\rm\bf (D)} The function $\phi_0(\cdot)=\phi_0(\cdot,c_1, c_2)$ is positive on $(c_1,c_2)$, moreover
$$
\forall \ee>0\quad \inf\{\phi_0(x,c_1, c_2):c_1-\ee<x<c_2-\ee\}>0.
$$
In addition, the function $\phi_0(\cdot,c_1, \infty)$ is continuous on $[c_1,\infty)$.

{\rm\bf (E)} functions $ \{\phi_k(\cdot)=\phi_k(\cdot,c_1, c_2)\} $ 
 form  a complete orthonormal system in the space  $L^2_{c_1,c_2}$ 
of all square-summable (with respect to the measure $\mu(y)dy$)
 functions on $(c_1,c_2)$; in particular, for all  $j>k\ge0$
$$
%\forall j>k\ge0\quad 
\int_{c_1}^{c_2}\phi_k^2(y,c_1, c_2)\mu(y) dy=1
\quad \mbox{and} \quad 
\int_{c_1}^{c_2}\phi_k(y,c_1, c_2)\phi_j(y,c_1, c_2)\mu(y) dy=0.
$$
\end{lemma}

\subsection{Corollaries from Lemma \ref{L1}} 
Under  conditions \eqref{i5} consider the func\-tion:
\begin{gather}                                                \label{j4}
r_{c_1,c_2}(a,T,y) :=   T^{\lambda_0} q_{c_1,c_2}(a,T,y) -\phi_0(a) \phi_0(y) 
= T^{\lambda_0} \sum\limits_{k=1}^{\infty} \frac{\phi_k(a) \phi_k(y)}{T^{\lambda_k}}
\end{gather}
with the same
functions $ \{\phi_k(\cdot)\} $ and  numbers $\{\lambda_k\}$ 
as in  \eqref{j3}.

\begin{lemma}                                      \label{L2}
Suppose that conditions \eqref{i5} and \eqref{i24} are fulfilled.
Then 
\begin{align}                                                \label{j5}
T^{2(\lambda_1-\lambda_0)}r_{c_1,c_2}^2(a,T,y) &\le (u_av_y)^{2(\lambda_1-\lambda_0)} r_{c_1,c_2}(a,u_a^2,a)r_{c_1,c_2}(y,v_y^2,y)
\\                                                                  \label{j6}
& \le (u_av_y)^{2\lambda_1} q_{c_1,c_2}(a,u_a^2,a)q_{c_1,c_2}(y,v_y^2,y).
\end{align}
\end{lemma}

\begin{proof}
Note first of all that by Schwartz inequality
\begin{align}                                                       \label{j7}
 \Sigma:=   \sum\limits_{k=1}^{\infty} \frac{|\phi_k(a)|}{u_a^{\lambda_k}} \frac{|\phi_k(y)|}{v_y^{\lambda_k}}
   &\le \sqrt{\sum\limits_{k=1}^{\infty} \frac{\phi_k^2(a)}{u_a^{2\lambda_k}}  }
    \sqrt{\sum\limits_{k=1}^{\infty} \frac{\phi_k^2(y)}{v_y^{2\lambda_k}}}  
    \\ \nonumber
    &=\sqrt{ \frac{r(a,u_a^2,a)}{u_a^{2\lambda_0}}}  
    \sqrt{ \frac{r(y,v_y^2,y)}{v_y^{2\lambda_0}}} . 
\end{align}
Here and below we use simplified notation $r(\cdot,\cdot,\cdot)$ instead of $r_{c_1,c_2}(\cdot,\cdot,\cdot)$.

On the other hand, we have  from assumptions \eqref{i24} that  for all $k\ge1$
\begin{gather*}             %                                        \label{j7+}
 \frac{1}{T^{\lambda_k}}= \frac{1}{T^{\lambda_1}} \frac{1}{T^{\lambda_k-\lambda_1}}\le \frac{1}{T^{\lambda_1}} \frac{1}{(u_av_y)^{\lambda_k-\lambda_1}}=\frac{(u_av_y)^{\lambda_1}}{T^{\lambda_1}} \frac{1}{(u_av_y)^{\lambda_k}}.
\end{gather*}
This elementary relation together with the definition of $ r(a,T,y)$ in  \eqref{j4} imply that
\begin{gather}                                                   \label{j8}
      \frac{|r(a,T,y)|}{T^{\lambda_0}}\le \sum\limits_{k=1}^{\infty} \frac{|\phi_k(a) \phi_k(y)|}{T^{\lambda_k}}\le \frac{(u_av_y)^{\lambda_1}}{T^{\lambda_1}} \sum\limits_{k=1}^{\infty} \frac{|\phi_k(a) \phi_k(y)|}{(u_av_y)^{\lambda_k}}
         =\frac{(u_av_y)^{\lambda_1}}{T^{\lambda_1}} \Sigma. 
\end{gather}
Substituting now  \eqref{j7} into  \eqref{j8} we arrive at \eqref{j5}.

Next, it follows from representations \eqref{j3} and \eqref{j4} that
\begin{gather*}       %                                            \label{j8+}
      \frac{r_{c_1,c_2}(y,T,y)}{T^{\lambda_0}}=\sum\limits_{k=1}^{\infty} \frac{\phi_k^2(y)}{T^{\lambda_k}}\le \sum\limits_{k=0}^{\infty} \frac{\phi_k^2(y)}{T^{\lambda_k}}=q_{c_1,c_2}(y,T,y) .
   \end{gather*}
   Using this inequality with $T=v_y^2$ and with $T=u_a^2$, when $y=a$, we obtain~\eqref{j6} as a consequence of  \eqref{j5}. 
\end{proof}


For all $x\in\R$ and $ t>0$ denote by  $\varphi (t,x) := \frac{1}{\sqrt{2\pi t}}\exp \left( -\frac{x^2}{2t}\right) $
 the density of $B_t$. 
 
 \begin{lemma}                                      \label{L3}
Under conditions \eqref{i5} 
\begin{align}                                                \label{j9}
   q_{c_1,c_2}(a,T,y)\le  \varphi (1-1/T,y-a/\sqrt T)\big/\mu(y).
\end{align}
In particular, when $a=y$ and $t=v^2$,
\begin{align}                                                \label{j10}
  \frac{\phi_0^2(y)}{v^{2\lambda_0}}\le
   q_{c_1,c_2}(y,v^2,y)\le  \frac{\varphi (1-1/v^2,y-y/v)}{\mu(y)}= \frac{  \varphi^2_v(y)}{\sqrt{2\pi}},
\end{align}
where $\varphi_v(y)$ was defined in  \eqref{i23}.
\end{lemma}

\begin{proof}
 
 
Note that $\varphi (1-1/T,\cdot-a/\sqrt T)$  is the density of $U_T=B_T/\sqrt{T}$ under condition that $B_1=a$.
Hence, under assumptions \eqref{i5}, we have from  \eqref{j2} and  \eqref{j1} that
\begin{align}                                                   \label{j11}
 \int_A  q_{c_1,c_2}(a,T,y) \mu(y) dy&\le\pr (\omega_{\log T} \in A | \omega_0 = a)
\\                                                              \nonumber
=\pr (U_T \in A | B_1 = a)&=\int_A   \varphi (1-1/T,y-a/\sqrt T) dy,
    \end{align}
and, as a result,  \eqref{j9} follows for each $y\in(c_1,c_2)$ because \eqref{j11} holds for all sets 
$A\subset(c_1,c_2)$.

Next, for $v>1$
$$
\frac{y^2}2-\frac{(y-y/v)^2}{2(1-1/v^2)}=\frac{y^2}2-\frac{y^2(v-1)^2}{2(v^2-1)}
=\frac{y^2(v+1)}{2(v+1)}-\frac{y^2(v-1)}{2(v+1)}=\frac{y^2}{v+1}.
$$
Hence, in this case
 \begin{gather*} %                                               \label{j11}
\frac{\varphi (1-1/v^2,y-y/v)}{\mu(y)}
    = \frac{1}{\sqrt{2\pi(1-1/v^2)}} \exp\left( \frac{y^2}{v+1}\right) = \frac{1}{\sqrt{2\pi}}\varphi^2_v(y).
\end{gather*}
So,  \eqref{j10} is also proved with  $\varphi_v(y)$  introduced in  \eqref{i23}.
\end{proof}

\subsection{Proof of Theorem \ref{T-Z} }
Below in the paper we use the next notations:
\begin{align}                                                \label{j12}
&\lambda:= \lambda(c_1,c_2):=\lambda_0(c_1,c_2)=\lambda_0
<\Lambda:=  \Lambda(c_1,c_2):=\lambda_1(c_1,c_2)=\lambda_1,
 \\                    \nonumber
 &\kk:=  \kk(c_1,c_2):=\lambda_1- \lambda_0>0,
 \quad 
 \theta_0(y,c_1,c_2):=\int_{c_1}^{y}\phi_0(x,c_1,c_2) \mu(x)dx.
  \end{align}
Note also that by \eqref{j1}, \eqref{j2}, and \eqref{i3} 
\begin{align}                                                   \label{j13}
P_{c_1,c_2}(a,T,y)=\int_{c_1}^{y} q_{c_1,c_2}(a,T,x) \mu(x)dx \qquad\text{on}\ \ \D.
\end{align}

 \begin{lemma}                                      \label{L4}
Suppose that the following equality
 \begin{align}                                                   \label{j14}
\Psi_{c_1,c_2}(a,y)\:=\psi_{c_1,c_2}(a)\theta_{c_1,c_2}(y)
=\Psi_0(a,y,,c_1,c_2):=\phi_0(a,c_1,c_2) \theta_0(y,c_1,c_2)
\end{align}
holds in domain  $\D$. Then estimate \eqref{i25} and convergence \eqref{i7} take place.
In particular, condition \eqref{j14} is obviously
 fulfilled when
$\psi_{c_1,c_2}(a)=\phi_0(a,c_1,c_2)$.
\end{lemma}

\begin{proof}
If condition \eqref{j14} is true, then we see from   \eqref{i11}, \eqref{j4},  and \eqref{j13} that
\begin{align}                                                   \label{j15}
\Delta_{c_1,c_2}(a,T,y)=\int_{c_1}^{y} r_{c_1,c_2}(a,T,x) \mu(x)dx.
\end{align}
Using now \eqref{j6}, \eqref{j10}, and \eqref{j12}  we get
\begin{align*}
|r_{c_1,c_2}(a,T,y)| 
	 \le \frac{(u_av_y)^{\lambda_1} }{T^{\lambda_1-\lambda_0}}
\sqrt{q_{c_1,c_2}(a,u_a^2,a)q_{c_1,c_2}(y,v_y^2,y)}
\\
 \le \frac{(u_av_y)^{\lambda_1} }{T^{\lambda_1-\lambda_0}}
\frac{  \varphi_{u_a}(a) \varphi_{v_y}(y)}{\sqrt{2\pi}}
=\frac{(u_av_y)^{\Lambda(c_1,c_2) }}{T^{\kk(c_1,c_2)}}
\frac{  \varphi_{u_a}(a) \varphi_{v_y}(y)}{\sqrt{2\pi}}.
\end{align*}
Plugging the last inequality into \eqref{j15} 
we arrive at \eqref{i25}.

At last, %note that
 convergence \eqref{i7} follows from estimate \eqref{i25} with $u_a=v_y=2$. 
\end{proof}

 \begin{lemma}                                      \label{L5}
Convergence \eqref{i7} takes place if and only if condition \eqref{j14} is fulfilled. 
In addition,   \eqref{j14} holds  if and only if there exists a constant $C>0$, independent of $a$ and $y$, such that 
\begin{align}                                                \label{j17}
 \psi_{c_1,c_2}(a)=C\phi_0(a,c_1,c_2) \ \quad \text{on}\ \ \D.
\end{align}
In particular, formula  \eqref{i30} with $C_{c_1,c_2}=C/\theta_0(y,c_1,c_2)$ immediately follows from  \eqref{j17} and  \eqref{j14}.
\end{lemma}

\begin{proof}
When $\psi_{c_1,c_2}=\phi_0$, convergence \eqref{i7} takes place by Lemma \ref{L4} with $\Psi_{c_1,c_2}(a,y)=\Psi_0(a,y,c_1,c_2)$. 
So, this equality holds for all functions $\psi_{c_1,c_2}$ which is possible to use in  \eqref{i7}. But condition $\Psi_{c_1,c_2}(a,y)=\Psi_0(a,y,c_1,c_2)$ coincides with  \eqref{j14}. 

Next, 
assume that condition \eqref{j14} is fulfilled. Recall that $||\phi_0||^2:=1$ as it follows from assertion (E) of Lemma 1. Hence in this case we have from  \eqref{i8} that
\begin{align*}      %                                             \label{j15+}
||\psi_{c_1,c_2}||^2=C^2>0
\quad\text{and}\quad
\theta_{c_1,c_2}(y)
:=\int_{c_1}^{y} \frac{C\phi_0(x,c_1,c_2)}{C^2}\mu(x)dx.
\end{align*}
Substituting these equalities into  \eqref{j14} we see that condition \eqref{j14} holds because $C^2/C^2=1$ and the product $\psi_{c_1,c_2}(a)\theta_{c_1,c_2}(y)$ is independent of $C>0$.

On the other hand, when  condition \eqref{j14} holds
with some possible function $\psi_{c_1,c_2}$, we have from \eqref{j14} with $y=c_2$ that identity \eqref{j17} takes place with
 \begin{align*}        %                                           \label{j16+}
C=\int_{c_1}^{c_2}\phi_0(x,c_1,c_2) \mu(x)dx\Big/\theta_{c_1,c_2}(c_2)>0.
\end{align*}

\end{proof}

So, there are no cases in Theorem \ref{T-Z} when condition \eqref{j14} does not hold. Hence,  Theorem \ref{T-Z} is proved in Lemma \ref{L4} in all cases.

\subsection{Proof of   Theorem \ref{TT-Z}}
 From  \eqref{i6} and  \eqref{j10} with $v=v_y$ and $v=u_a$ we immediately have that
 % on $\D$
  \begin{align}                                                \label{j22}
0\le\phi_0(y)\le
   \frac{ v_y^\lambda \varphi_{v_y}(y)}{\sqrt[4]{2\pi}}
   \quad\text{and}\quad
   0\le\phi_0(a)\le
   \frac{ u_a^\lambda \varphi_{u_a}(a)}{\sqrt[4]{2\pi}}
   \qquad\text{on}\ \ \D.
\end{align}
Now, by definition  \eqref{j12},
 \begin{align}                                                   \label{j23}
0\le\theta_0(y,c_1,c_2)=\int_{c_1}^{y}\phi_0(x) \mu(x)dx
\le\int_{c_1}^{y}\frac{ v_x^\lambda \varphi_{v_x}(x)}{\sqrt[4]{2\pi}}\mu(x)dx.
\end{align}
Substituting  \eqref{j22} and  \eqref{j23} into the representation  \eqref{j14} for 
$\Psi_{c_1,c_2}(a,y) $
we arrive at  \eqref{i28}.

At last, %note that  
\eqref{i26} is evident by  \eqref{i11}, because the probability
\mbox{$P_{c_1,c_2}(a,T,y) \le1$}. So, both assertions of Theorem \ref{TT-Z} are proved.






\subsection{Two auxiliary lemmas}
First of all note that for $v>1$
\begin{align}                                                                 \label{j25}
\beta_2(v):=\sqrt[4]{1-1/v^2}=\sqrt[4]{\frac{v^2}{v^2-1}}
=\sqrt[4]{\frac{v^2}{(v-1)(v+1)}}<\beta_1(v):=\sqrt[4]{\frac{v+1}{v-1}}.
%\quad v>1.
\end{align}

\begin{lemma}                                      \label{L6}
For all $y\ge c_1$
\begin{align}                                                                 \label{j26}
\oo\theta_v(y):=\int_{c_1}^y\ff_{v}(x)\mu(x)dx\le\sqrt{2\pi}\beta_1(v),
\quad v>1.
\end{align}
\end{lemma}

\begin{proof}
From definition  \eqref{i23} we see that
\begin{align*}      %                                                           \label{z2}
\ff_{v}(y)\mu(y)=\beta_2(v) 
\exp\left( \frac{y^2}{2(v+1)}-\frac{y^2}{2} \right)
=\beta_2(v) \exp\left( -\frac{y^2}{2\sigma^2(v)} \right)\hspace{-1mm},
\quad v>1.
\end{align*}
with $\sigma^2(v):=(v+1)/v$. Hence
\begin{align}                                                                 \label{j27}
\oo\theta_v(y)=\int_{c_1}^y\ff_{v}(x)\mu(x)dx\le\beta_2(v) 
\int_{-\infty}^\infty\exp\left( -\frac{x^2}{2\sigma^2(v)} \right)dx
=\sqrt{2\pi}\beta_2(v)\sigma(v)
\end{align}
for $v>1$, where
\begin{align*}  %                                                               \label{z4}
\beta_2(v)\sigma(v)=\sqrt[4]{\frac{v^2}{v^2-1}}
=\sqrt[4]{\frac{v^2}{(v-1)(v+1)}\cdot\frac{(v+1)^2}{v^2}}
=\sqrt[4]{\frac{v+1}{v-1}}=\beta_1(v).
%\quad v>1.
\end{align*}
Thus,  \eqref{j26} follows from  \eqref{j27}.
\end{proof}

For $i=1,2$ introduce functions:
\begin{align}                                                                 \label{j28}
f_1(v):=\beta_1(v)v^\Lambda
\quad\text{and}\quad
f_2(v,y):=\beta_2(v)v^\Lambda\ff_v(y),
\quad v>1,\quad y\in\R,
\end{align}
where the value $\Lambda=\lambda_1$ was defined in \eqref{j12}. Note that
\begin{align*}                %                                                 \label{j29}
0<\kk<\Lambda<\infty
\quad\text{and}\quad
1<v_*:=1+\frac1{4\Lambda}<1+\frac1{\kk}<\infty.
\end{align*}

\begin{lemma}                                      \label{L7}
For all $y\in\R$ and $K\ge1$
\begin{align}                                                                 \label{j30}
f_2(v,y)\le \sqrt[4]{e(1+8\Lambda)}K^\Lambda\exp\left( \frac{y^2}{2v} \right)
\quad\text{when}\quad v_*\le v\le v_*K.
\end{align}
In addition, for all $y\ge c_1$
\begin{align}                                                                 \label{j31}
v_*^\Lambda\oo\theta_{v_*}(y)\le v_*^\Lambda\sqrt{2\pi}\beta_1(v_*)
=\sqrt{2\pi}f_1(v_*)
\le\sqrt{2\pi}\sqrt[4]{e(1+8\Lambda)}.
\end{align}
\end{lemma}

\begin{proof}
From definitions  \eqref{j25} and conditions  \eqref{j30} on number $v$ we have:
\begin{align*}           %                                                      \label{j25}
\beta^4_2(v)<\beta^4_1(v)=1+\frac2{v-1}\le1+\frac2{v_*-1}=1+8\Lambda,
\\
v^\Lambda\le v_*^\Lambda K^\Lambda
=\left( 1+\frac1{4\Lambda} \right)^\Lambda K^\Lambda
\le e^{1/4}K^\Lambda=\sqrt[4]{e}K^\Lambda.
\end{align*}               %
Substituting this estimates into definition  \eqref{j28} of functions $f_i$, we arrive at~\eqref{j30}.
When $K=1$, these arguments and \eqref{j26} also imply   \eqref{j31}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proof of   Theorem \ref{T-1} }
We are going to show that under assumption~\eqref{i14} 
\begin{align}                                                \label{k1}
\forall\, T>1\ \ \forall u_a>1\quad\big|\Delta_{c_1,c_2}(a,T,y)\big| \le \frac{f_2(u_a,a)f_1(v_*)}{ T^{\kk}} 
\qquad\text{on}\ \ \D.
\end{align}
First of all we apply  Theorem \ref{T-Z} with $v_y=v_*$.  In this case the main condition~\eqref{i24} of this theorem has the form:
\begin{align}                                                        \label{k2}
  u_a>1\qquad \mbox{and} \qquad T\ge u_av_*,
%  \qquad\text{on}\ \ \D,
\end{align}
and assertion \eqref{i25} may be rewritten in the following way:
\begin{align}                                                \label{k3}
\big|\Delta_{c_1,c_2}(a,T,y)\big| \le \frac{u_a^{\Lambda}\ff_{u_a}(a)}{\sqrt{2\pi} T^{\kk}}v_*^{\Lambda}\theta_{v_*}(y) 
 =\frac{f_2(u_a,a)}{\sqrt{2\pi} T^{\kk}}  v_*^{\Lambda}\theta_{v_*}(y) 
\quad\text{on}\ \ \D,
\end{align}
where we used simplified notations, introduced in \eqref{i23}, \eqref{j26},  and \eqref{j28}.
 But now \eqref{k1} immediately follows from \eqref{k3} and \eqref{j31}.
 
 Thus, we proved \eqref{k1} when assumption \eqref{k2} is true. Now suppose that~\eqref{k2} is not fulfilled and assume instead that the next condition takes place:
\begin{align}                                                        \label{k4}
  u_a>1\qquad \mbox{and} \qquad1< T\le u_av_*.
%  \qquad\text{on}\ \ \D.
\end{align}
 Now we will apply Theorem \ref{TT-Z}.
 From the first inequality in \eqref{i26}, using \eqref{k4}, we get:
 \begin{align}                                                \label{k5}
\Delta_{c_1,c_2}(a,T,y) \le T^\lambda=\frac{ T^\Lambda}{ T^\kk}
&\le\frac{ (u_av_*)^\Lambda}{ T^\kk}
\\ \nonumber
&\le\frac{ (u_av_*)^\Lambda}{ T^\kk}
\ff_{u_a}(a)\beta_1(v_*)=\frac{ f_2(u_a,a)f_1(v_*)}{ T^\kk}
%\qquad\text{on}\ \ \D,
\end{align}
on $\D$, because $\ff_u(a)>1$ and $\beta_1(v)>1$.

On the another hand, under assumption \eqref{k4} we find from \eqref{i26} and \eqref{i28} that 
\begin{align*}                     %                           \label{i26}
-\Delta_{c_1,c_2}(a,T,y)&\le \Psi_{c_1,c_2}(a,y)\le \frac{ (u_av_*)^\kk}{ T^\kk}\Psi_{c_1,c_2}(a,y)
\\       %    \label{i28}
 &\le \frac{ (u_av_*)^\kk}{ T^\kk}\frac{u_a^{\lambda}\ff_{u_a}(a)}{\sqrt{2\pi} }
v_*^{\lambda}\oo\theta_{v_*}(y)=\frac{u_a^{\Lambda}\ff_{u_a}(a)}{T^\kk\sqrt{2\pi} }v_*^{\Lambda}\oo\theta_{v_*}(y)
\quad\text{on}\ \D.
\end{align*}
Using now notations from  \eqref{j28} and estimate  \eqref{j31} we obtain:
\begin{align*}                     %                           \label{i26}
-\Delta_{c_1,c_2}(a,T,y)\le 
\frac{f_2(u_a,a)}{T^\kk\sqrt{2\pi} }v_*^{\Lambda}\oo\theta_{v_*}(y)
\le \frac{f_2(u_a,a)}{T^\kk}f_1(v_*)
\quad\text{on}\ \ \D.
\end{align*}
Thus, it follows from the last inequality and \eqref{k5} that estimate \eqref{k1} is true under assumption \eqref{k4}. 

So, \eqref{k1} is proved in all cases. But the desired estimate \eqref{i15} in  Theorem~\ref{T-1} follows immediately from \eqref{k1} and \eqref{j31}. Hence,  Theorem \ref{T-1} is also proved. 

\subsection{Proof of   Corollary \ref{C-2}}
We are going to use inequality \eqref{j30} for $a$ instead of $y$ and $u_a$ instead of $v$, where 
\begin{align*}                   %                                        \label{j30}
u_a=v_*K,  \quad\text{with}\quad K=(1+b_aa^2),\ b_a>0.
\end{align*}
Then $a^2/u_a\le a^2/K\le 1/b_a$. Hence, by \eqref{j30},
\begin{align*}             %                                                    \label{j30}
f_2(u_a,a)\le \sqrt[4]{e(1+8\Lambda)}(1+b_aa^2)^\Lambda\exp\left( \frac1{2b_a} \right)
\quad\text{on}\ \D.
\end{align*}
Substituting this estimate into \eqref{i15}, we obtain:
\begin{align*}             %                                   \label{i18}
\big|\Delta_{c,\infty}(a,T,y)\big| &\le \sqrt[4]{e(1+8\Lambda)}\frac{f_2(u_a,a)}{ T^{\kk}} 
\\
&\le\sqrt[4]{e(1+8\Lambda)}\frac{ \sqrt[4]{e(1+8\Lambda)}(1+b_aa^2)^\Lambda}{ T^{\kk}}\exp\left( \frac1{2b_a} \right)
\ \ \text{on} \ \D.
\end{align*}
It is easy to see that the last inequality coincides with \eqref{i18}.

Thus, Corollary \ref{C-2} is true. 
Hence, all the results of the paper are proved.

\vspace{2mm}
{\bf Acknowledgment.} The author is thankful to the referee for careful reading
of the paper and very useful comments.






   
\begin{thebibliography}{1}

   
    \bibitem{Breiman65} L. Breiman, 
  \textcolor{blue}   {\it First exit times from a square root boundary,}
     Proc. Fifth Berkeley Symp. Math. Stat. Prob., {\bf 2} (1965), 1236--1249.


    \bibitem{DHSW22} D. Denisov, G. Hinrichs, A.I. Sakhanenko and V. Wachtel,
\textcolor{blue} {\it Crossing an asymptotically square root boundary by the Brownian motion,}
     Proceedings of the Steklov Institute of Mathematics, {\bf 316} (2022), 105--120.
    
\bibitem{DSW16} D. Denisov, A.I. Sakhanenko,   and V. Wachtel, 
\textcolor{blue} {\it First-passage times for random walks with non-identically distributed increments,}
{Ann. Probab.}, {\bf 46}:6 (2018), 3313--3350.

\bibitem{Gaertner82} J. G\"artner, 
\textcolor{blue} {\it Location of wave fronts for the multi-mimensional KPP equation and Brownian first exit densities,}
 Math. Nachr., {\bf 105} (1982), 317--351.
    
    \bibitem{GP83} P. Greenwood,  and E. Perkins, 
\textcolor{blue} {\it A conditioned limit theorem for random walk and Brownian local time on square root boundaries,}
    {Ann. Probab.}, \textbf{11} (1983), 227--261.

\bibitem{Novikov79} A.A. Novikov, 
\textcolor{blue} {\it On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary,}
{\em Mathematics of the USSR -- Sbornik}, 
{\bf 38}:4 (1981), 495-505.


\bibitem{Novikov81} A.A. Novikov, 
\textcolor{blue} {\it Martingale approach to first passage problems of nonlinear boundaries,}
 {Proc. Steklov Inst. Math.}, {\bf 158} (1981), 130--152.

\bibitem{Sato77} Sh. Sato, 
\textcolor{blue} {\it Evaluation of the first-passage time probability to a square root boundary for the Wiener process,}
 {J. Appl. Probab.}, {\bf 14} (1977), 850--856.

\bibitem{U80} K. Uchiyama,  
\textcolor{blue} { \it Brownian first exit from and sojourn over one-sided moving boundary and application,}
 { Z. Wahrsch. Verw. Gebiete}, {\bf 54} (1980), 75-116.

\end{thebibliography}
   
    \end{document}

