Some limit theorems for the critical Galton–Watson branching processes
Abstract
UDC 519.21 We consider critical Galton–Watson processes starting from a random number of particles and determine the effect of the mean value of the initial state on the asymptotic state of the process. For processes starting from a large number of particles and satisfying the condition <mml:math> <mml:mrow> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> we prove the limit theorem similar to the result of W. Feller. We also prove the theorem under the condition <mml:math> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> for critical processes satisfying the conditions <mml:math> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> </mml:math> and <mml:math> <mml:mrow> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:math>