Linearity of regression for weak records, revisited

Since many years characterization of distribution by linearity of regression of non-adjacent weak records E(W_{i+s}|W_i) = \beta_1 W_i+\beta_0 for discrete observations has been known to be a difficult question. Lopez- Blazquez (2004) proposed an interesting idea of reducing it to the adjacent case and claimed to have the characterization problem completely solved. We will explain that, unfortunately, there is a major aw in the proof given in that paper. This aw is related to fact that in some situations the operator responsible for reduction of the non-adjacent case to the adjacent one is not injective. The operator is trivially injective when 0<\beta_1<1. We show that when \beta_1>=1 the operator is injective when s = 2, 3, 4. Therefore in these cases the method proposed by Lopez-Blazquez is valid. We also show that the operator is not injective when \beta_1 >=1 and s >= 5. Consequently, in this case the reduction methodology does not work and thus the characterization problem remains open.