BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Right-Most Position of a Last Progeny Modified Branching Random Wa
 lk - Antar Bandyopadhyay\, ISI Delhi
DTSTART:20231004T150000Z
DTEND:20231004T160000Z
UID:TALK206554@talks.cam.ac.uk
CONTACT:Sourav Sarkar
DESCRIPTION:In this talk\, we will consider a modification of the usual Br
 anching Random Walk (BRW)\, where we will give certain independent and ide
 ntically distributed  (i.i.d.) displacements/perturbations to all the part
 icles at the generation $n$. We call this process last progeny modified br
 anching random walk (LPM-BRW). Depending on the value of a parameter\, $\\
 theta > 0$\, which works as a "scale parameter'' for the perturbations\, w
 e will classify the model in three distinct cases\, namely\, the boundary 
 case\, below the boundary case\, and above the boundary case. Under very m
 inimal assumptions on the underlying point process of the increments\, we 
 will show that in the boundary case\, the maximum displacement converges t
 o a limit after only an appropriate centering\, which will be the form $c_
 1 n - c_2 \\log n$. We will give explicit formulas for the constants $c_1$
  and $c_2$ and will show that $c_1$ is exactly the same\, while $c_2$ is $
 1/3$ of the corresponding constants of the Classical BRW [Aídekon 2013]. 
 We will also be able to characterize the limiting distribution as a random
 ly shifted Gamble distribution. We will further show that below the bounda
 ry the logarithmic correction term will be absent\, while for above the bo
 undary case\, the logarithmic correction term is  exactly same as that of 
 the classical BRW. If time permits then we will also show that Brunet-Derr
 ida-type results of point process convergence of our LPM-BRW to a Poisson 
 point process also hold. Our proofs are based on a novel method of couplin
 g the maximum displacement with a linear statistic associated with a well-
 studied process in statistics known as the smoothing transformation.\n\n[T
 his is a joint work with Partha Pratim Ghosh]
LOCATION:MR11
END:VEVENT
END:VCALENDAR