We consider a sequence of random length M of independent absolutely continuous
observations Xi, 1 ≤ i ≤ M, where M is geometric, X1 has cdf G, and Xi, i ≥ 2, have
cdf F. Let N be the number of upper records and Rn, n ≥ 1, be the nth record value.
We show that N is free of F if and only if G(x) = G0(F (x)) for some cdf G0 and that if
E(|X2|) is finite so is E(|Rn|) for n ≥ 2 whenever N ≥ n or N = n. We prove that the
distribution of N along with appropriately chosen subsequences of E(Rn) characterize
F and G, and along with subsequences of E(Rn − Rn−1) characterize F and G up to a
common location shift. We discuss some applications to the identification of the wage
offer distribution in job search models.