Truncated Moments for Heavy-Tailed and Related Distribution Classes - Saulius Paukštys; Jonas Šiaulys; Remigijus Leipus (Mathematics, 2023)
Suppose that <semantics>ξ+</semantics> is the positive part of a random variable defined on the probability space <semantics>(Ω,F,P)</semantics> with the distribution function <semantics>Fξ</semantics>. When the moment <semantics>E<mfenced open="(" close=")">ξ+</mfenced>p</semantics> of order <semantics>p>0</semantics> is finite, then the truncated moment <semantics><mover>F¯</mover>ξ,p(x)=min<mfenced separators="" open="{" close="}">1,E<mfenced separators="" open="(" close=")">ξp1<mspace width="-2.9pt"></mspace>I{ξ>x}</mfenced></mfenced></semantics>, defined for all <semantics>x⩾0</semantics>, is the survival function or, in other words, the distribution tail of the distribution function <semantics>Fξ,p</semantics>. In this paper, we examine which regularity properties transfer from the distribution function <semantics>Fξ</semantics> to the distribution function <semantics>Fξ,p</semantics> and which properties transfer from the function <semantics>Fξ,p</semantics> to the function <semantics>Fξ</semantics>. The construction of the distribution function <semantics>Fξ,p</semantics> describes the truncated moment transformation of the initial distribution function <semantics>Fξ</semantics>. Our results show that the subclasses of heavy-tailed distributions, such as regularly varying, dominatedly varying, consistently varying and long-tailed distribution classes, are closed under this truncated moment transformation. We also show that exponential-like-tailed and generalized long-tailed distribution classes, which contain both heavy- and light-tailed distributions, are also closed under the truncated moment transformation. On the other hand, we demonstrate that regularly varying and exponential-like-tailed distribution classes also admit inverse transformation closures, i.e., from the condition that <semantics>Fξ,p</semantics> belongs to one of these classes, it follows that <semantics>Fξ</semantics> also belongs to the corresponding class. In general, the obtained results complement the known closure properties of distribution regularity classes.