  In many physical, social or economical phenomena we observe changes of a
studied quantity only in discrete, irregularly distributed points in time. The
stochastic process used by physicists to describe this kind of variables is the
Continuous Time Random Walk (CTRW). Despite the popularity of this type of
stochastic processes and strong empirical motivation, models with a long-term
memory within the sequence of time intervals between observations are missing.
Here, we fill this gap by introducing a new family of CTRWs. The memory is
introduced to the model by the assumption that many consecutive time intervals
can be the same. Surprisingly, in this process we can observe a slowly decaying
nonlinear autocorrelation function without a fat-tailed distribution of time
intervals. Our model applied to high-frequency stock market data can
successfully describe the slope of decay of nonlinear autocorrelation function
of stock market returns. The model achieves this result with no dependence
between consecutive price changes. It proves the crucial role of inter-event
times in the volatility clustering phenomenon observed in all stock markets.
