Krämer (Sankhy<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\bar{\mathrm{a }}$$</EquationSource> </InlineEquation> 42:130–131, <CitationRef CitationID="CR13">1980</CitationRef>) posed the following problem: “Which are the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathbf{y}$$</EquationSource> </InlineEquation>, given <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\mathbf{X}$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathbf{V}$$</EquationSource> </InlineEquation>, such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\mathbf{y}$$</EquationSource> </InlineEquation>...</equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></citationref></equationsource></inlineequation>

We show that the distribution of the product and the sum of independent random variables having arithmetic distributions is again arithmetic. In case of the product, it is also possible to give a formula for the span. Furthermore, we also prove the converse statement. If the distribution of the...