On a class of random walks in simplexes

We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. From an interior point $z$, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on $z$, and then the particle randomly jumps to a new location $z'$ on the segment connecting $z$ to the chosen vertex. In some specific cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are, in fact, Dirichlet. We also consider a related history-dependent random walk model in $[0,1]$ based on an urn-type scheme. We show that this random walk converges in distribution to the arcsine law.Comment: final versio

Classical solution of the Cauchy problem for biwave equation: Application of Fourier transform

In this paper, we use some Fourier analysis techniques to find an exact solution to the Cauchy problem for the $n$-dimensional biwave equation in the upper half-space $\mathbb{R}^n\times [0,+\infty)$.Comment: 11 page

Early diagnosis and prognosis of severe dengue

Dengue is considered as an emerging infectious disease with a wide expansion to more than 100 countries and territories in the world. Dengue has great social, economic and health impacts on endemic regions causing the disease burden to the community and becoming one of the most serious public health concerns. We described clinical and virological features of paediatric dengue cases in southern Vietnam in a large prospective cohort study. We used the new classification of WHO 2009 to include dengue shock syndrome, severe bleeding and organ failure in the category of severe dengue in data analysis. Of 7544 patients enrolled into the study with complete haemobiochemical results available, 2060 patients (27.3%) had laboratory- confirmed diagnosis of dengue. The number of cases with severe dengue, dengue requiring parenteral fluids and uncomplicated dengue was 117 (5.7%), 156 (7.6%) and 1787 (86.7%) respectively. Proportions of RT-PCR positivity in the groups of severe dengue, dengue requiring parenteral fluids and uncomplicated dengue was 115/117 (98.3%), 149/156 (95.5%) and 1690/1787 (94.6%) respectively. It is a challenge for attending physicians in the outpatient settings to recognize early dengue patients. A simple tool, called Early Dengue Classifier (EDC), including information of age, white blood cell and platelet counts can be used alone or in combination with the NS1 rapid test to make early diagnosis of dengue within 72 hours of onset illness. We demonstrate that the early diagnosis of dengue can be enhanced beyond the current standard of care using a simple evidence-based algorithm. A prognostic algorithm using vomiting, platelet count, aspartate aminotransferase and NS1 rapid test result, called Early Severe Dengue Identifier (ESDI), can predict severe complications. Though the positive predictive value of the ESDI was low as seen with common prognostic models for a rare outcome, the results should support patient management and clinical trials of specific therapies

Malliavin-Stein method for multi-dimensional U-statistics of Poisson point processes

In this paper, we give an upper bound for a probabilistic distance between a Gaussian vector and a vector of U-statistics of Poisson point processes by applying Malliavin-Stein inequality on the Poisson space.Comment: 11 page

Speed of excited random walks with long backward steps

We study a model of multi-excited random walk with non-nearest neighbour steps on $\mathbb Z$, in which the walk can jump from a vertex $x$ to either $x+1$ or $x-i$ with $i\in \{1,2,\dots,L\}$, $L\ge 1$. We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton-Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh [Probab. Theory Related Fields (2008), 141 (3-4)], we extend their result (w.r.t the case $L=1$) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift $\delta>2$. This confirms a special case of a conjecture proposed by Davis and Peterson.Comment: 27 page