# Downsample a stochastic process without losing correlation statistics?

I have a stochastic variable $$X(t)$$ which changes at a discrete set of random times $$t_1, t_2, \dots$$. I can simulate this stochastic process to obtain a series $$X(t_1), X(t_2),\dots$$

However, the transitions are so common that saving the entire simulation is too memory intensive. My first approach was to only store the state of the process $$(t_i, X(t_i))$$ if $$t_i-t_k> \Delta t$$, where $$t_k$$ was the last time at which the state was saved. So $$\Delta t$$ is a sampling interval and $$1/\Delta t$$ is a sampling rate. However, downsampling in this way, only recording the state of the system about every $$\Delta t$$, biases the statistics of transition times. For example, it's difficult to estimate the distribution of the time between transitions, i.e., of $$\delta t = t_i-t_{i-1}$$ if I downsample in this way, since I ignored all $$\delta t < \Delta t$$.

Is there a way to downsample a stochastic time series that (even approximately) preserves the statistics of transition times? Please note I have to downsample on the fly, i.e., as the simulation runs. So I wonder if there is a random way to save the state of the process which generates an approximately unbiased statistical sample, preserving the relative number of transitions at each timescale.