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A continuous-state branching process with immigration having branching mechanism
and immigration mechanism
, a CBI
process for short, may have either of two different asymptotic regimes, depending on whether
, the CBI process has either a limit distribution or a growth rate dictated by the branching dynamics. When
, immigration overwhelms branching dynamics. Asymptotics in the latter case are studied via a nonlinear time-dependent renormalization in law. Three regimes of weak convergence are exhibited. Processes with critical branching mechanisms subject to a regular variation assumption are studied. This article proves and extends results stated by M. Pinsky in ‘Limit theorems for continuous state branching processes with immigration’ (Bull. Amer. Math. Soc.78, 1972).
Numerous studies have confirmed the rapid retreat of Tibetan Plateau glaciers in recent decades, and resulting reductions in glacier volume. However, high-resolution determinations of the changes in glacier thickness remain sparse. This paper presents results based on differential GPS measurements to accurately measure glacier thickness change over the past few years. Measurements from the lower part of Gurenhekou glacier show an average thickness change of –3.82 m over a 4 year period. On the lower part of Kangwure glacier we measured an average thickness change of –2.70 m over 3 years. On the upper part of Naimona’Nyi glacier (northern branch), western Himalaya, thickness changed by –1.34 m on average between 2008 and 2010, and –0.87 m between 2010 and 2013. Large temporal changes in thinning rates were found on Naimona’Nyi glacier, due to variations in local precipitation. Our measurements also show variable changes in glacier thickness over different parts of each glacier, with little dependence on elevation. The limited data also show glacier thinning in the accumulation zone.
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