  Possible distributions are discussed for intertrade durations and
first-passage processes in financial markets. The view-point of renewal theory
is assumed. In order to represent market data with relatively long durations,
two types of distributions are used, namely, a distribution derived from the
so-called Mittag-Leffler survival function and the Weibull distribution. For
Mittag-Leffler type distribution, the average waiting time (residual life time)
is strongly dependent on the choice of a cut-off parameter t_ max, whereas the
results based on the Weibull distribution do not depend on such a cut-off.
Therefore, a Weibull distribution is more convenient than a Mittag-Leffler type
one if one wishes to evaluate relevant statistics such as average waiting time
in financial markets with long durations. On the other side, we find that the
Gini index is rather independent of the cut-off parameter. Based on the above
considerations, we propose a good candidate for describing the distribution of
first-passage time in a market: The Weibull distribution with a power-law tail.
This distribution compensates the gap between theoretical and empirical results
much more efficiently than a simple Weibull distribution. We also give a useful
formula to determine an optimal crossover point minimizing the difference
between the empirical average waiting time and the one predicted from renewal
theory. Moreover, we discuss the limitation of our distributions by applying
our distribution to the analysis of the BTP future and calculating the average
waiting time. We find that our distribution is applicable as long as durations
follow a Weibull-law for short times and do not have too heavy a tail.
