\begin{table}%t1
\caption{\label{tab1}Slope $s$ of a power law fit ($P\propto t_{\rm p}^{-s}$)
of the~$t_{\rm p}$ -- $P$~diagram with
and without the assumption of the DLR (Eq.~(8)) for pulses.}
\par
%\centering
\par
\begin{tabular}{ccc}
\hline \hline
\noalign{\smallskip}
$L_{\rm min}$ (erg~s$^{-1}$) & $s$ (with DLR) 
& $s$ (without DLR)\\
\hline
$10^{51}$ & 0.27 & 0.091 \\
$10^{50.5}$ & 0.24 & 0.093 \\
$10^{50}$ & 0.22 & 0.096 \\
\hline 
$L_{\rm max}$ (erg~s$^{-1}$) & 
& \\
$10^{53.5}$ & 0.24 & 0.093 \\
$10^{54}$ & 0.22 & 0.096 \\
\hline  
$\delta$ & & \\ 
 2.0 & 0.27 & 0.081 \\
 1.5 & 0.27 & 0.096 \\
\hline
$E_{\rm p}$ (keV) & & \\
400 & 0.33 & 0.089 \\
600 & 0.33 & 0.088 \\
800 & 0.33 & 0.096 \\
\hline 
$\sigma_{\rm t}$ & & \\
0.3 & 0.27 & -- \\
0.6 & 0.22 & -- \\
1.0 & 0.15 & -- \\
1.5 & 0.09 & -- \\
\hline
Threshold & & \\
:2 & 0.30 & 0.11 \\
x2 & 0.24 & 0.085 \\
\hline
\end{tabular}
\tablefoot {In the six blocks
we respectively vary the lower (i); and upper (ii) limits of the pulse luminosity
function; 
(iii) the slope of the luminosity function; (iv) the central 
value of a log-normal distribution for $E_{\rm p}$; (v) the dispersion
of the DLR; (vi) the detection threshold.  The first row corresponds to
our reference case with $L_{\rm min}=10^{51}$~erg~s $^{-1}$,
$L_{\rm max}=10^{53}$~erg~s$^{-1}$,
$\delta=1.7$, $\sigma_{\rm t}=0.3$~dex. It also assumes the validity of the
Amati-like relation (Eq.~(7)) with a dispersion of 0.3~dex and adopts the
threshold criterion for BATSE given by Band (\cite{Band03}). In each block only
one parameter is varied, the others keeping the values corresponding to
the reference case.}
\end{table}