Max-Compound Cox Processes. I

Extreme values are considered in samples with random size that have a mixed Poisson distribution that is generated by a doubly stochastic Poisson process. Some inequalities are proved relating the distributions and moments of extrema with those of the leading process (the mixing distribution). Limit theorems are proved for the distributions of max-compound Cox processes, and limit distributions are described. An important particular case of the negative binomial distribution of a sample size corresponding to the case where the Cox process is led by a gamma Lévy process is considered, explaining a possible genesis of tempered asymptotic models.