Large deviations for sums defined on a Galton-Watson process

In this paper we study the large deviation behavior of sums of i.i.d. random variables X_i defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of the partial sums of the X_i and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06]

Lower deviation probabilities for supercritical Galton-Watson processes

There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With 'lower deviation probabilities' we refer to P(Z_n=k_n) with k_n=o(c_n) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Z_{n+1}/Z_n. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramer method for proving large deviations of sums of independent variables to our needs

The Joint IGNTP/INTF Editio Critica Maior of the Gospel of John: its goals and their significance for New Testament scholarship

Conference paper delivered at SNTS Annual Meeting in Halle, Germany, August 200

Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma, where gamma is an index associated to the medium.Comment: 40 page

Critical Galton-Watson processes: The maximum of total progenies within a large window

Consider a critical Galton-Watson process Z={Z_n: n=0,1,...} of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m is such that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_{infinity}(j).Comment: 28 page

On the left tail asymptotics for the limit law of supercritical Galton--Watson processes in the Böttcher case

Under a well-known scaling, supercritical Galton-Watson processes $Z$ converge to a non-degenerate non-negative random limit variable $W.$ We are dealing with the left tail (i.e. lose to the origin) asymptotics of its law. In the Bötcher case (i.e. if always at least two offspring are born), we describe the precise asymptotics exposing tiny oscillations (Theorem 1). Under a reasonable additional assumption, the oscillations disappear (Corollary 2). Also in the B"ottcher case, we improve a recent lower deviation probability result by describing the precise asymptotics under a logarithmic scaling (Theorem 3). Under additional assumptions, we even get the fine (i.e. without log-scaling) asymptotics (Theorem 4)

Large scale localization of a spatial version of Neveu’s branching process

AbstractRecently a spatial version of Neveu’s (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process

Large scale localization of a spatial version of Neveu's branching process

Recently a spatial versions of Neveu's (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process

MPTP-induced degeneration: interference with glutamatergic toxicity

Parkinson's disease (PD) is characterised by the progressive degeneration of nigrostriatal dopamine (DA) neurons resulting in the major symptoms of akinesia and rigidity. Although the primary cause of PD is still not known some features make this disorder a model for neurodegenerative diseases in general. It has been known for some time that symptomatic PD can be attributed to insults with symptoms occurring many years later such as post-encephalitic PD or PD following manganese poisoning. More recently, MPTP (1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine) has been identified as a neurotoxin selective for melanin-containing dopaminergic neurons in humans and non-human primates. The specificity of this neurotoxin and the striking clinical similarities to idiopathic PD, seen in primates, make MPTP-induced parkinsonism the most useful animal model of a neurological disease. There are numerous theoretical possibilities to interfere with both MPTP-induced neurotoxicity and the symptomatology of PD. In recent years excitatory amino acids have gained considerable interest since they can cause excitotoxic lesion of neurons under a number of pathological conditions (Olney et al., 1989; Choi, 1988). Here we summarise the present data and provide new experimental evidence indicating that MPTP-induced degeneration of dopaminergic neurons does involve glutamate-mediated toxicity. It is concluded that glutamate-mediated excitotoxicity results in the destruction of DAergic somata in the substantia nigra. Non-competitive or competitive NMDA antagonists protect nigral neurons from MPTP-induced degeneration whereas their striatal terminals still seem to degenerate

Einige Bemerkungen zum senatorischen cursus honorum vor dem Konsulat

Avtor analizira službeni potek (cursus honorum) legatov onih legij, ki so bile stacionirane v vzhodnih provincah rimskega imperija. Pri tem ugotavlja za senatorske kariere pred konsulatom naslednje. L Zdi se, da do preture nobena služba nima bistvenega vpliva na poznejši službeni potek. Opaziti je samo, da so bili legijski legati že pred to službo v milosti pri vladarju. Mnogi so bili kvestorji, ljudski tribuni, edili ali pretorji kot candidati Augusti ali pa so opravljali službo ab actis senatus. 2. Med preturo in legijskim poveljstvom pridejo v poštev predvsem službe — navadno ena ali dve — legatus proconsulis, iuridicus, praefectus frumenti dandi, curator viae. Niso pa opravljali pred legijskim poveljstvom sledečih služb: namestništvo v pretorskih cesarskih provincah in prefektura cesarske blagajne. Tudi namestništvo v pretorskih senatorskih provincah, ki ga najdemo v dveh primerih, se zdi izjemno. 3. Neposredno napredovanje od legijskega poveljstva v konsulat je redko in nastopi le pri izrednih zaslugah. Če je med legijskim poveljstvom in konsulatom le ena služba, je normalno to namestništvo v cesarski provinci z eno legijo. Vse druge službe so izjemne, kot se zdi. Navadno sta med legijskim poveljstvom in konsulatom dve službi. Za ti dve pridejo v poštev sledeče skupine uradnih mest: guvernerska mesta v cesarskih provincah brez legije, prokonsulati in blagajniške prefekture, in sicer iz posamezne skupine le po ena služba. 4. Izvor legatov vzhodnih legij se sklada s splošnimi razvojnimi tendencami v sestavi rimskega senata. Že v prvi polovici 1. stoletja n. e. nastopajo nekateri homines novi iz Italije, proti koncu Neronove vlade prispeva prve provincialce Gallia Narbonensis in Hispania Baetica, prvi orijentalec se pojavi pod Flavijci, prvi Afrikanci pa okoli srede 2. stoletja. Od tedaj provincialci prevladujejo, vendar tudi še v 3. stoletju Italikov ne izpodrinejo povsem