Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment

This is a short survey about asymptotic properties of a supercritical branching process <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>Z</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Zₙ) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>W</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (Wₙ) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> Lᵖ for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> p≥ 1 , and the boundedness of the harmonic moments <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="double-struck">E</m:mi> <m:mo>⁢</m:mo> <m:msubsup> <m:mi>W</m:mi> <m:mi>n</m:mi> <m:mrow> <m:mo>-</m:mo> <m:mi>a</m:mi> </m:mrow> </m:msubsup> </m:mrow> </m:math> EWₙ⁻ᵃ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> a>0 . We then present limit theorems and large deviation results on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>log</m:mi> <m:mo>⁡</m:mo> <m:msub> <m:mi>Z</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:math> log Zₙ , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.