SOME CHARACTERIZATIONS OF EXPONENTIAL OR GEOMETRIC DISTRIBUTIONS IN ANONSTATIONARY RECORD VALUE MODEL

Let (F(n)}n greater-than-or-equal-to 0 be a sequence of c.d.f. and let {R(n)}n greater-than-or-equal-to 1 be the sequence of record values i n a non-stationary record model where after the (n - 1)th record the p opulation is distributed according to F(n). Then the equidistribution of the nth population and the record increment R(n) - R(n - 1) (i.e. R (n) - R(n - 1) approximately F(n)) characterizes F(n). to have an expo nentially decreasing hazard function. To be more precise F(n) is the e xponential distribution if the support of R(n - 1) generates a dense s ubgroup in R and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be Z, any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chose n arbitrarily. Setting lambda = - log(1 - F(1)), F(n)(x) = 1 - F(x - [ x])exp(- lambda[x]) for-all x greater-than-or-equal-to 0 is an admissi ble solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that F(n) is either exponential or geometric are given. Similar results for expo nential or geometric tail distributions based on the independence of R (n - 1) and R(n) - R(n - 1) are proved.