Suppose X(t) = minn≤N(t) Y n , for t ≥ 0, where N = n δT n is a Poisson process on R+ and Y n are i.i.d. with distribution F, independent of N. For instance, Yn could be bids on a property and X(t) is...

Suppose X(t) = minn≤N(t) Y_n, for t ≥ 0, where N = n δT_n
is a Poisson process on R+ and Y_n
are i.i.d. with distribution F, independent of N. For instance, Yn could be bids on a property and X(t) is the smallest bid up to time t. Find the distribution and mean of X(t). Answer this question for the more general setting in which Y_n
are p-marks of Tn, where p(t,(0, y]) is the distribution of a typical mark at time t.