| Summary: | Assume that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> are two independent random variables with distribution functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>η</mi></msub></semantics></math></inline-formula>, respectively. The distribution of a random variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mi>η</mi></mrow></semantics></math></inline-formula>, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>ξ</mi></msub><mo>⊗</mo><msub><mi>F</mi><mi>η</mi></msub></mrow></semantics></math></inline-formula>, is called the product-convolution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>η</mi></msub></semantics></math></inline-formula>. It is proved that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>ξ</mi></msub><mo>⊗</mo><msub><mi>F</mi><mi>η</mi></msub></mrow></semantics></math></inline-formula> is a generalized subexponential distribution if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mi>ξ</mi></msub></semantics></math></inline-formula> belongs to the class of generalized subexponential distributions and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is nonnegative and not degenerated at zero.
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