Title: Linear processes generated by independent random variables
Abstract: In a recent paper R. Dudley gave a characterization of those sequences of independent and identically distributed random variables which are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{l_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-compatible for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-over-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≧<!-- ≧ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p \geqq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the present note we extend his result into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-parenthesis 0 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p \in (0,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and provide some conditions (necessary or sufficient) for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript phi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>φ<!-- φ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{l_\varphi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-compatibility of a sequence of independent random variables not necessarily identically distributed.