Convergence in Total Variation of Random Sums

• Main Authors: Luca Pratelli, Pietro Rigo
• Format: Article
• Language: English
• Published: MDPI AG 2021-01-01
• Series: Mathematics
• Subjects: exchangeability; random sum; rate of convergence; stable convergence; total variation distance
• Online Access: https://www.mdpi.com/2227-7390/9/2/194

Let <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> be a sequence of real random variables, <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>T</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> a sequence of random indices, and <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>τ</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> a sequence of constants such that <inline-formula><math display="inline"><semantics><mrow><msub><mi>τ</mi><mi>n</mi></msub><mo>→</mo><mi>∞</mi></mrow></semantics></math></inline-formula>. The asymptotic behavior of <inline-formula><math display="inline"><semantics><mrow><msub><mi>L</mi><mi>n</mi></msub><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msub><mi>τ</mi><mi>n</mi></msub><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>T</mi><mi>n</mi></msub></msubsup><msub><mi>X</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, as <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></semantics></math></inline-formula>, is investigated when <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is exchangeable and independent of <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>T</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula>. We give conditions for <inline-formula><math display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mo>=</mo><msqrt><msub><mi>τ</mi><mi>n</mi></msub></msqrt><mspace width="0.166667em"></mspace><mrow><mo>(</mo><msub><mi>L</mi><mi>n</mi></msub><mo>−</mo><mi>L</mi><mo>)</mo></mrow><mo>⟶</mo><mi>M</mi></mrow></semantics></math></inline-formula> in distribution, where <i>L</i> and <i>M</i> are suitable random variables. Moreover, when <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> is i.i.d., we find constants <inline-formula><math display="inline"><semantics><msub><mi>a</mi><mi>n</mi></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>b</mi><mi>n</mi></msub></semantics></math></inline-formula> such that <inline-formula><math display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">sup</mo><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="script">B</mi><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></msub><mspace width="0.166667em"></mspace><mrow><mo>|</mo><mrow><mi>P</mi><mrow><mo>(</mo><msub><mi>L</mi><mi>n</mi></msub><mo>∈</mo><mi>A</mi><mo>)</mo></mrow><mo>−</mo><mi>P</mi><mrow><mo>(</mo><mi>L</mi><mo>∈</mo><mi>A</mi><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>≤</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">sup</mo><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="script">B</mi><mo>(</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></msub><mspace width="0.166667em"></mspace><mrow><mo>|</mo><mrow><mi>P</mi><mrow><mo>(</mo><msub><mi>M</mi><mi>n</mi></msub><mo>∈</mo><mi>A</mi><mo>)</mo></mrow><mo>−</mo><mi>P</mi><mrow><mo>(</mo><mi>M</mi><mo>∈</mo><mi>A</mi><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>≤</mo><msub><mi>b</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> for every <i>n</i>. In particular, <inline-formula><math display="inline"><semantics><mrow><msub><mi>L</mi><mi>n</mi></msub><mo>→</mo><mi>L</mi></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><msub><mi>M</mi><mi>n</mi></msub><mo>→</mo><mi>M</mi></mrow></semantics></math></inline-formula> in total variation distance provided <inline-formula><math display="inline"><semantics><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math display="inline"><semantics><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow></semantics></math></inline-formula>, as it happens in some situations.