Similar agreement will be kept for other functions which involve $\tau (n)+R,$** **$\tau (nt)+R$** or** $\tau (nt,n)+R.$

\[min2\] ([@VD3], Corollary 3) Assume that $A1-$ $A2$ hold. For any $t\in \left( 0,1\right] $ and $\varepsilon >0$ $$\limsup_{n\rightarrow \infty }\mathbf{P}\left( \,\alpha
_{n}(nt)>1+\varepsilon \,\left\vert \,\tau \left( n\right) <nt\right.
\right) =0.$$

The next statement complements Lemmas \[min1\] and \[min2\].

\[Lprospect1\] Assume that $A1-$ $A2$ hold. Then for $R\in \mathbb{Z},$ any $t\in \left( 0,1\right] ,$ and any $\varepsilon >0$$$\limsup_{R,n\rightarrow \infty }\mathbf{P}\left( \alpha _{n}(\tau
(nt,n)+R)>1+\varepsilon \left\vert \,\tau \left( n\right) <nt\right. \right)
=0.$$

**Proof**. First we note that  $\alpha _{n}(m)\downarrow 1$ as $m\uparrow n$. Hence $$\begin{aligned}
\alpha _{n}(\tau (nt,n)+R)I\left\{ nt\leq \tau (nt,n)+R\leq n\right\}  &\leq
&\alpha _{n}(nt)I\left\{ nt\leq \tau (nt,n)+R\leq n\right\}  \\
&\leq &\alpha _{n}(nt).\end{aligned}$$Therefore, $$\begin{aligned}
\mathbf{P}\left( \alpha _{n}(\tau (nt,n)+R)>1+\varepsilon ;\tau \left(
n\right) <nt\right)  &\leq &\mathbf{P}\left( \alpha _{n}(nt)>1+\varepsilon
;\tau \left( n\right) <nt\right)   \notag \\
&&+\mathbf{P}\left( \tau (nt,n)+R\notin \lbrack nt,n]\right) .  \label{Adr1}\end{aligned}$$Observing that $\left\{ \tau (nt,n)+R>n\right\} =\oslash $ for $R<0$ and $\left\{ \tau (nt,n)+R<nt\right\} =\oslash $ for $R>0$ and recalling the generalized arcsine law (which, under Assumption $A1,$ holds for $n^{-1}\tau
(nt,n)$ (see [@S])) we see that for each fixed $R$ $$\lim_{n\rightarrow \infty }\mathbf{P}\left( \tau (nt,n)+R\notin \lbrack
nt,n]\right) =0.  \label{Interm}$$Passing now to the limit as $n\rightarrow \infty $ in 