An urn model for cannibal behavior

A sampling procedure involving an urn with red and white balls in it is studied. Initially, the urn contains n balls, r of them being white. At each step, a white ball is removed, and one more ball is selected at random, painted red (if it was white before) and put back into the urn. R. F. Green proposed this scheme in 1980 as a stochastic model of cannibalistic behavior in a biological population, with red balls interpreted as cannibals. Of primary interest is the distribution of Xnr, the terminal number of red balls. A study of R. F. Green and C. A. Robertson led them to conjecture that, for r = 1 and n →∞, Xnr is asymptotically normal with mean ≈ n exp(–1) and variance ≈ n(3 exp(–2) –exp(–1)). In this paper we prove that the conjecture — its natural extension, in fact — is true. Namely, for r/n bounded away from 1, Xnr is shown to be asymptotically normal with mean ≈ n exp(ρ – 1) and variance ≈ n exp[2(ρ – 1)] (ρ2– 3ρ + 3 – exp(l – ρ)); ρ = r/n.

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Reduced Branching Processes with Very Heavy Tails

Journal of Applied Probability ◽

10.1239/jap/1208358961 ◽

2008 ◽

Vol 45 (1) ◽

pp. 190-200 ◽

Cited By ~ 4

Author(s):

Andreas N. Lagerås ◽

Serik Sagitov

Keyword(s):

Stochastic Model ◽

Branching Process ◽

Critical Case ◽

Heavy Tails ◽

Regularity Condition ◽

Branching Processes ◽

Time Interval ◽

Biological Population ◽

Long Time ◽

Heavy Tailed

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

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On a memory game and preferential attachment graphs

Advances in Applied Probability ◽

10.1017/apr.2016.16 ◽

2016 ◽

Vol 48 (2) ◽

pp. 585-609 ◽

Cited By ~ 2

Author(s):

Hüseyin Acan ◽

Paweł Hitczenko

Keyword(s):

Preferential Attachment ◽

Random Variable ◽

Urn Model ◽

Limiting Distributions ◽

Asymptotically Normal ◽

Generalized Pólya Urn ◽

Limit Result ◽

Few Vertices ◽

Memory Game ◽

Main Technical Tool

Abstract In their recent paper Velleman and Warrington (2013) analyzed the expected values of some of the parameters in a memory game; namely, the length of the game, the waiting time for the first match, and the number of lucky moves. In this paper we continue this direction of investigation and obtain the limiting distributions of those parameters. More specifically, we prove that when suitably normalized, these quantities converge in distribution to a normal, Rayleigh, and Poisson random variable, respectively. We also make a connection between the memory game and one of the models of preferential attachment graphs. In particular, as a by-product of our methods, we obtain the joint asymptotic normality of the degree counts in the preferential attachment graphs. Furthermore, we obtain simpler proofs (although without rate of convergence) of some of the results of Peköz et al. (2014) on the joint limiting distributions of the degrees of the first few vertices in preferential attachment graphs. In order to prove that the length of the game is asymptotically normal, our main technical tool is a limit result for the joint distribution of the number of balls in a multitype generalized Pólya urn model.

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Reduced Branching Processes with Very Heavy Tails

Journal of Applied Probability ◽

10.1017/s0021900200004058 ◽

2008 ◽

Vol 45 (01) ◽

pp. 190-200

Author(s):

Andreas N. Lagerås ◽

Serik Sagitov

Keyword(s):

Stochastic Model ◽

Branching Process ◽

Critical Case ◽

Heavy Tails ◽

Regularity Condition ◽

Branching Processes ◽

Time Interval ◽

Biological Population ◽

Long Time ◽

Heavy Tailed

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

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A model for an epidemic occurring in a number of isolated communities

Advances in Applied Probability ◽

10.1017/s0001867800040659 ◽

1975 ◽

Vol 7 (3) ◽

pp. 462-462

Author(s):

Tony Woods

Keyword(s):

Markov Chain ◽

Stochastic Model ◽

Classical Model ◽

Natural Extension ◽

Control Policy ◽

Embedded Markov Chain ◽

Threshold Behaviour ◽

Isolated Communities

The model is just the natural extension to n dimensions of the classical model for an epidemic with removals: however, the deterministic version does show a richer variety of threshold behaviour. In the stochastic model, some results will be discussed for the embedded Markov chain, and also a possible control policy involving the removal of entire communities.

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High-resolution electron microscopy of Cu2O surfaces

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100143584 ◽

1986 ◽

Vol 44 ◽

pp. 396-397

Author(s):

V. Castano ◽

W. Krakow

Keyword(s):

High Resolution ◽

Natural Extension ◽

High Resolution Electron Microscopy ◽

Point Of View ◽

Initial Oxidation ◽

System A ◽

Resolution Electron ◽

Surface Reconstructions ◽

Atomic Surface ◽

Au Thin Films

In non-UHV microscope environments atomic surface structure has been observed for flat-on for various orientations of Au thin films and edge-on for columns of atoms in small particles. The problem of oxidation of surfaces has only recently been reported from the point of view of high resolution microscopy revealing surface reconstructions for the Ag2O system. A natural extension of these initial oxidation studies is to explore other materials areas which are technologically more significant such as that of Cu2O, which will now be described.

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Opening images of synaptic vesicles and calcium localization by means of microwave fixation method

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100134508 ◽

1990 ◽

Vol 48 (2) ◽

pp. 182-183

Author(s):

Vinci Mizuhira ◽

Hiroshi Hasegawa

Keyword(s):

Synaptic Vesicles ◽

Room Temperature ◽

Osmium Tetroxide ◽

Sampling Procedure ◽

Fixation Method ◽

Potassium Oxalate ◽

X Ray ◽

Cacodylate Buffer ◽

Ultrathin Sections ◽

Microwave Fixation

Microwave irradiation (MWI) was applied to 0.3 to 1 cm3 blocks of rat central nervous system at 2.45 GHz/500W for about 20 sec in a fixative, at room temperature. Fixative composed of 2% paraformaldehyde, 0.5% glutaraldehyde in 0.1 M cacodylate buffer at pH 7.4, also contained 2 mM of CaCl2 , 1 mM of MgCl2, and 0.1% of tannic acid for conventional observation; and fuether 30-90 mM of potassium oxalate containing fixative was applied for the detection of calcium ion localization in cells. Tissue blocks were left in the same fixative for 30 to 180 min after MWI at room temperature, then proceeded to the sampling procedure, after postfixed with osmium tetroxide, embedded in Epon. Ultrathin sections were double stained with an useal manner. Oxalate treated sections were devided in two, stained and unstained one. The later oxalate treated unstained sections were analyzed with electron probe X-ray microanalyzer, the EDAX-PU-9800, at 40 KV accelerating voltage for 100 to 200 sec with point or selected area analyzing methods.

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Universal ESEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149763 ◽

1993 ◽

Vol 51 ◽

pp. 786-787

Author(s):

G.D. Danilatos

Keyword(s):

Electron Microscope ◽

Scanning Electron Microscope ◽

High Vacuum ◽

Natural Extension ◽

Necessary Condition ◽

Environmental Scanning ◽

Detection Systems ◽

Small Gain ◽

Detection Medium ◽

Scanning Electron

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.

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