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Doubly uniform complete law of large numbers for independent point processes

We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for the random variables taking values in functions of $2$ parameters which share similar monotonicity properties as the increments of monotone functions. The assumptions for the main result are the Hölder continuity on the expectations as well as moment conditions, while the sample functions may contain jumps. In particular, we can apply the results to point processes (counting processes) which lack Markov or martingale type properties.