We know that merging two Poisson processes results in another Poisson process with a rate that is the sum of the two original rates. (https://www.probabilitycourse.com/chapter11/11_1_3_merging_and_splitting_poisson_processes.php)
What type of process do we get by merging two processes with lognormal (interarrival time) distributions? How are the parameters of this process related to the parameters of the original lognormal processes?
The main difference is that a Poisson process is a Markov process, which is not the case of a process with lognormal interarrival time distributions.
Suppose that your process 1 has just jumped. The question is : what is the next time of jump of one of the two processes ? The answer heavily depends on the past of process 2: for instance, suppose the variance of your processes is very small, and the mean is 1. At time 0, the process 1 has just jumped, and if we look at the past we see that the last jump of process 2 occured at -x, 0
The distribution of a process that is the sum of two lognormal processes has no simple closed form answer. There are lots of approximations, the best so far seems to be the rather messy one described in https://arxiv.org/pdf/1502.03619.
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Guven, can you answer the question "How does one "merge" two lognormal processes?" specifically?
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