Tid: 28 maj 1998 kl 1515-1700 (Observera dagen och tiden!)

Plats : Seminarierummet 3733, Institutionen för matematik, KTH, Lindstedts väg 25, plan 7. Karta!

Föredragshållare: Malcolm Quine, School of Mathematics and Statistics, University of Sydney. (Publikationslista)

Titel: Asymptotic Normality of Processes with a Branching Structure

Summary with link to complete paper


Let tex2html_wrap_inline36 be integer-valued rv's defined on the same probability space tex2html_wrap_inline38 and define sequences tex2html_wrap_inline40 and tex2html_wrap_inline42 by tex2html_wrap_inline44 , tex2html_wrap_inline46 , tex2html_wrap_inline48 and


where tex2html_wrap_inline52 denotes the indicator of tex2html_wrap_inline54 .

The process tex2html_wrap_inline56 contains the classical Bienaymé-Galton-Watson branching process as a special case, and is also relevant to urn models and to a `self-annihilating branching process' considered by Erickson (Ann. Probab. 1973) as a model for the antigenic behaviour of Lymphoma cell populations.

We will first review the results of Quine and Szczotka (Ann. Appl. Probab. 1994) concerning the extinction probability tex2html_wrap_inline58 and the asymptotic behaviour of tex2html_wrap_inline60 and tex2html_wrap_inline62 , when tex2html_wrap_inline64 , including

1) the a.s. convergence of tex2html_wrap_inline66 to some non-negative rv tex2html_wrap_inline68 as tex2html_wrap_inline70 , for some constant tex2html_wrap_inline72 and

2) when tex2html_wrap_inline74 , rate of convergence results of the form




where tex2html_wrap_inline80 is a standard normal rv independent of tex2html_wrap_inline82 , for some tex2html_wrap_inline84 .

We will then discuss further unpublished joint work with Szczotka, investigating the extent to which tex2html_wrap_inline60 and tex2html_wrap_inline62 share similar asymptotic properties. For instance we consider the analogues for the sequence tex2html_wrap_inline90 , and establish an equivalence relation between these convergences. We show that in some situations the rate is Gaussian or mixed Gaussian. We also investigate the asymptotic behaviour of tex2html_wrap_inline92 (or tex2html_wrap_inline94 ) when the centering involves the limit rv tex2html_wrap_inline68 .

Some of the results we obtain in this paper are known in the B-G-W case. However, our approach here is far more general: the summands tex2html_wrap_inline98 need be neither independent nor identically distributed (nor even non-negative). Our approach uses probabilistic methods based on an underlying Wiener process rather than methods based on p.g.f.'s or characteristic functions.

Till seminarielistan
To the list of seminars