On the prospective minimum of the random walk conditioned to stay nonnegative

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable columnalign="center" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:msub> <m:mi>S</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:msub> <m:mi>S</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mi>X</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mo>…</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>X</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mspace width="thinmathspace"/> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle S_{0}=0,\quad S_{n}=X_{1}+\ldots+X_{n},\, n\geq 1, \end{array}$$ be a random walk whose increments belong without centering to the domain of attraction of a stable law with scaling constants a n that provide convergence as n → ∞ of the distributions of the sequence { S n / a n , n = 1, 2, …} to this stable law. Let L r , n = min r ≤ m ≤ n S m be the minimum of the random walk on the interval [ r , n ]. It is shown that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mtable columnalign="center" rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:munder> <m:mo movablelimits="true">lim</m:mo> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munder> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi mathvariant="bold">P</m:mi> </m:mrow> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>y</m:mi> <m:msub> <m:mi>a</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow class="MJX-TeXAtom-ORD"> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:msub> <m:mi>S</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>t</m:mi> <m:msub> <m:mi>a</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>L</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mspace width="thinmathspace"/> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:mfenced> <m:mo>,</m:mo> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> $$\begin{array}{} \displaystyle \lim_{r,k,n\rightarrow \infty }\mathbf{P}\left( L_{r,n}\leq ya_{k}|S_{n}\leq ta_{k},L_{0,n}\geq 0\