  Energy markets and the associated energy futures markets play a crucial role
in global economies. We investigate the statistical properties of the
recurrence intervals of daily volatility time series of four NYMEX energy
futures, which are defined as the waiting times $\tau$ between consecutive
volatilities exceeding a given threshold $q$. We find that the recurrence
intervals are distributed as a stretched exponential $P_q(\tau)\sim
e^{(a\tau)^{-\gamma}}$, where the exponent $\gamma$ decreases with increasing
$q$, and there is no scaling behavior in the distributions for different
thresholds $q$ after the recurrence intervals are scaled with the mean
recurrence interval $\bar\tau$. These findings are significant under the
Kolmogorov-Smirnov test and the Cram{\'e}r-von Mises test. We show that
empirical estimations are in nice agreement with the numerical integration
results for the occurrence probability $W_q(\Delta{t}|t)$ of a next event above
the threshold $q$ within a (short) time interval after an elapsed time $t$ from
the last event above $q$. We also investigate the memory effects of the
recurrence intervals. It is found that the conditional distributions of large
and small recurrence intervals differ from each other and the conditional mean
of the recurrence intervals scales as a power law of the preceding interval
$\bar\tau(\tau_0)/\bar\tau \sim (\tau_0/\bar\tau)^\beta$, indicating that the
recurrence intervals have short-term correlations. Detrended fluctuation
analysis and detrending moving average analysis further uncover that the
recurrence intervals possess long-term correlations. We confirm that the
"clustering" of the volatility recurrence intervals is caused by the long-term
correlations well known to be present in the volatility.

Statistical Finance