<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13171_2025_399_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$X_1, X_2,\ldots $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> </mrow> </math></EquationSource> </InlineEquation> be independent non-negative integer-valued random variables and <i>N</i> a non-negative integer-valued random variable independent of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13171_2025_399_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$X_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>’s. We derive bounds in Poisson approximation for the random sum <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13171_2025_399_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$W_N=\sum _{i=1}^N X_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>N</mi> </msub> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msub> <mi>X</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, and the sum <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13171_2025_399_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">$W_n=\sum _{i=1}^n X_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>n</mi> </msub> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>X</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13171_2025_399_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}(N=n)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">P</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>=</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in stop-loss metrics of order 1 and 2 through Stein’s method and the zero bias transformation. As part of our applications, we provide specific bounds for the net stop-loss premium and the collateralized debt obligation.</p>

Poisson Approximation for Stop-Loss Metrics of Order 1 and 2

期刊论文
发表日期：

2025年7月21日

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Nat Yonghint,
Wasamon Jantai

摘要

Let $X_1, X_2,\ldots $ $X_{1}, X_{2}, \dots$ be independent non-negative integer-valued random variables and N a non-negative integer-valued random variable independent of $X_i$ $X_{i}$ ’s. We derive bounds in Poisson approximation for the random sum $W_N=\sum _{i=1}^N X_i$ $W_{N} = \sum_{i = 1}^{N} X_{i}$ , and the sum $W_n=\sum _{i=1}^n X_i$ $W_{n} = \sum_{i = 1}^{n} X_{i}$ when $\mathbb {P}(N=n)=1$ $P (N = n) = 1$ in stop-loss metrics of order 1 and 2 through Stein’s method and the zero bias transformation. As part of our applications, we provide specific bounds for the net stop-loss premium and the collateralized debt obligation.