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1

Using multitype branching models to analyze bacterial pathogenicity

Tahir Daniah, Kaj Ingemar, Bartoszek Krzysztof, Majchrzak Marta, Parniewski Paweł, Sakowski Sebastian

Mathematica Applicanda

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2020

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Vol. 48, no. 1

59--86

EN

We apply multitype continuous time Markov branching models to study pathogenicity in E. coli, a bacterium belonging to the genus Escherichia. First, we examine briefly the properties of multitype branching processes and we also survey some fundamental limit theorems regarding the behavior of such models under various conditions. These theorems are then applied to discrete, state dependent models in order to analyze pathogenicity in a published clinical data set consisting of 251 strains of E. coli. We use well established methods, incorporating maximum likelihood techniques, to estimate speciation rates as well as the rates of transition between diffrerent states of the models. From the analysis, we not only derive new results, but we also verify some preexisting notions about virulent behavior in bacterial strains.

PL

W celu zbadania patogenności szczepów E. coli, bakterii z rodzaju Escherichia, użyto wielorodzajowego markowskiego procesu gałązkowego z czasem ciągłym. W pierwszej kolejności zrobiono przegląd własności wykorzystywanego procesu wraz z najważniejszymi wynikami granicznymi opisującymi zachowanie procesu przy różnych założeniach. Następnie przyjęto konkretny model, w którym rozgałęzienia są zależne od stanu oraz zastosowano go do badania patogenności w zestawie 251 szczepów E. coli pochodzących z II Centralnego Szpitala Klinicznego Wojskowej Akademii Medycznej. Parametry modelu, tempa narodzin oraz mutacji, zostały uzyskane metodą największej wiarygodności. Przedstawiona w artykule analiza potwierdza znane własności patogenności bakterii oraz sugeruje nowe ścieżki pracy badawczej.

2

On a limit structure of the Galton-Watson branching processes with regularly varying generating functions

Imomov Azam A.

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

61--73

EN

We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type E [x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.

3

Asymptotic Behavior for Quadratic Variations of Non-Gaussian Multiparameter Hermite Random Fields

Tran T. T. Diu

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 2

385--401

EN

Let (Ztq,H)t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = (H1,…, Hd) ∈ (1/2, 1)d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion (q = 1, d = 1), fractional Brownian sheet (q = 1, d ≥ 2), the Rosenblatt process (q = 2, d = 1) as well as the Rosenblatt sweet (q = 2, d ≥ 2). For any q ≥ 2, d ≥ 1 and H ∈ (1/2, 1)d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2(Ω) to a standard d-parameter Rosenblatt random variable with self-similarity index Hʺ = 1 + (2H − 2)/q.

4

Maejima M., Tudor C. A.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 1

167--186

EN

We study selfsimilar processes with stationary increments in the second Wiener chaos. We show that, in contrast with the first Wiener chaos which contains only one such process (the fractional Brownian motion), there is an infinity of selfsimilar processes with stationary increments living in the Wiener chaos of order two. We prove some limit theorems which provide a mechanism to construct such processes.

5

On the rate of convergence in non-central asymptotics of the Hermite variations of fractional Brownian sheet

Breton J. -C

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 2

301--311

EN

The Hermite variations of the anisotropic fractional Brownian sheet enjoy similar behaviour to that for the fractional Brownian motion: central (convergence to a normal distribution) or non-central (convergence to a Hermite-type distribution). In this note, we investigate the rate of convergence in the non-central case.

6

q-analogs of order statistics

Kordecki W., Łyczkowska-Hanćkowiak A.

Probability and Mathematical Statistics

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2010

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Vol. 30, Fasc. 2

207--214

EN

We introduce the notion of the q-analog of the k-th order statistics. We give a distribution and asymptotic distributions of q-analogs of the k-th order statistics and the intermediate order statistics with r → ∞ and r − k → ∞ in the projective geometry PG (r − 1; q).

7

Random sums stopped by a rare event : a new approximation

Gamboa F., Pamphile P.

Probability and Mathematical Statistics

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2004

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Vol. 24, Fasc. 2

237--252

EN

The convergence of a geometric sum of positive i.i.d. random variables to an exponential distribution is a well-known result. This convergence provided various and useful approximations in reliability, queueing or risk theory. However, for concrete applications, this exponential approximation is not sharp enough for small values of mission time. So, other approximations have been proposed (Bon and Pamphile (2001), Kalashnikov (1997)). In this paper we propose a new point of view where the exponential approximation appears as a first-order approximation. We consider more general random sums stopped by a rare event, where summands are no more assumed to be independent neither nonnegative. So we give a second-order approximation. As illustration we consider stopping time with negative binomial distribution. This approximation provides a new evaluation tool in reliability analysis of highly reliable systems. The accuracy of this approximation is studied numerically.

8

The rate of convergence in the precise large deviation theorem

Baltrūnas A.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 2

343--354

EN

Let X1, X2, . . . be i.i.d. random variables with a common d.f. F. Let Sn = X1 +. . .+ Xn, n ≥1, and Mn = max k≤n Xk, n≥1. In this paper for a large class of subexponential distributions we estimate the rate of convergence. ∆n(t) = P(Sn > t) − P(Mn> t), where n ≥ 1 and t ≥ 0. We close this paper with some examples.

9

Sequences of iterates of random-valued vector functions and continuous solutions of a linear functional equation of infinite order

Kapica R.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 4

447-455

EN

Given a probability space (Ω, A, P), a separable Banach space X, and measurable functions L: Ω → (0, +∞), M: Ω → X we obtain some theorems on the existence and on the uniqueness of continuous solutions φ: X → R of the equation φ (x) = ∫Ω φ (L(ω) x + M (ω)) P (dω).

10

Strong limit theorems for general renewal processes

Klesov O., Rychlik Z., Steinebach J.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 2

329--349

EN

An approach is discussed to derive strong limit theorems for general renewal processes from the corresponding asymptotics of the underlying renewal sequence. Neither independence nor stationarity of increments is required. In certain situations, just the dualities between the renewal processes and their defining sequencesin combination with some regularity conditions on the normalizing constants are sufficient for the proofs. There are other cases, however, in which the duality argumentsdo not apply, and where other techniques have to be developed. Finally, there are also examples, in which an inversion of the limit theorems under consideration cannot work at all.