Christina Goldschmidt (Oxford Oxford University)
Title:听Globally centred snakes.
Abstract: We consider a globally centred discrete snake on a Bienaym茅 tree with critical, finite variance offspring distribution which satisfies a finite global variance condition. Making use of a discrete line-breaking construction from a recent paper of Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin, we give a new proof of convergence (under rescaling) to the Brownian snake driven by a Brownian excursion in the sense of finite-dimensional distributions. The conditions required for tightness are a work in progress, but we demonstrate the necessity of a natural tail-condition on the displacements. This extends earlier results of various authors including Janson and Marckert for the case where the displacements are independent of the offspring numbers, and Marckert for the globally centred, global finite variance case restricted to bounded offspring distributions. Our results apply, in particular, to give the convergence of a special discrete snake which gives the difference between the height process of such a Bienaym茅 tree and a constant multiple of its 艁ukasiewicz path, when rescaled by n^{-1/4}.
This is joint work in progress with Louigi Addario-Berry, Serte Donderwinkel and Rivka Mitchell.
听