On decomposition problem for distribution functions of class 𝑄

We consider a new class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper Q"> <mml:semantics> <mml:mi mathvariant="bold-italic">Q</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of distribution functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 1"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 2"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 1 equals upper F asterisk upper F 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mo> ∗ </mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">F_1=F*F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A distribution function of the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper Q"> <mml:semantics> <mml:mi mathvariant="bold-italic">Q</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy–Khinchine type representation with a “signed spectral measure”. The class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper Q"> <mml:semantics> <mml:mi mathvariant="bold-italic">Q</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , being a natural extension of the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper I"> <mml:semantics> <mml:mi mathvariant="bold-italic">I</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {I}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan, and Sato [Trans. Amer. Math. Soc. 370 (2018), pp. 8483–8520] formulated the open question: is it true that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F element-of bold-italic upper Q"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi mathvariant="bold-italic">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">F\in \boldsymbol {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F equals upper F 1 asterisk upper F 2"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ∗ </mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">F=F_1*F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with some distribution functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 1"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 2"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 1 element-of bold-italic upper Q"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mi mathvariant="bold-italic">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">F_1\in \boldsymbol {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 2 element-of bold-italic upper Q"> <mml:semantics> <mml:mrow>