Long-Time Behavior of Galton–Watson Systems with Circular Mechanism

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>Z</mi><mi>n</mi></msub><mo>:</mo><mi>n</mi><mo>≥</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> be a Galton–Watson system with a circular mechanism <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">a</mi><mo>∗</mo><mi mathvariant="bold-italic">b</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">a</mi><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mi>a</mi><mi>j</mi></msub><mo>}</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">b</mi><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mi>b</mi><mi>j</mi></msub><mo>}</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></msubsup></mrow></semantics></math></inline-formula> are probability distributions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="bold">Z</mi><mo>+</mo></msub><mo>:</mo><mo>=</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>a</mi></msub><mo>:</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></munderover></mstyle><mi>j</mi><msub><mi>a</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>b</mi></msub><mo>:</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mo>∞</mo></munderover></mstyle><mi>j</mi><msub><mi>b</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula>. The extinction property of such branching systems is first studied. Then, it is proved that there exists <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>Γ</mi><mi>n</mi></msub></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mi>n</mi></msub><mo>=</mo><msubsup><mi>Γ</mi><mi>n</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>Z</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> is an integrable martingale and hence converges to some random variable <i>W</i>. Moreover, for the case that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>1</mn></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the convergence rates to 0 of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><mfenced separators="" open="|" close="|"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>Z</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>Z</mi><mi>n</mi></msub></mfrac></mstyle><mo>−</mo><msub><mi>ω</mi><mi>n</mi></msub></mfenced><mo>></mo><mi>ε</mi><mo>∣</mo><msub><mi>Z</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mfenced><mrow><mo>,</mo><mi>P</mi><mo>(</mo><mo>|</mo></mrow><msub><mi>W</mi><mi>n</mi></msub><mo>−</mo><mi>W</mi><mrow><mo>|</mo><mo>></mo><mi>ε</mi><mo>∣</mo><msub><mi>Z</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mfenced separators="" open="(" close=")"><mfenced separators="" open="|" close="|"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>Z</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>Z</mi><mi>n</mi></msub></mfrac></mstyle><mo>−</mo><msub><mi>ω</mi><mi>n</mi></msub></mfenced><mo>></mo><mi>ε</mi><mo>∣</mo><mi>W</mi><mo>≥</mo><mi>δ</mi><mo>,</mo><msub><mi>Z</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn></mfenced></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula> are presented for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ε</mi><mo>,</mo><mi>δ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> under various moment conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>a</mi><mi>j</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>b</mi><mi>j</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>n</mi></msub><mo>=</mo><msub><mi>m</mi><mi>a</mi></msub></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>b</mi></msub></semantics></math></inline-formula> for <i>n</i> being even or odd, respectively. It is further shown that the first rate is geometric while the last two rates are supergeometric under a finite moment generating function hypothesis.