Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Large deviations for Generalized Polya Urns with arbitrary urn function   We consider a generalized two-color Polya urn (black and withe balls) first
introduced by Hill, Lane, Sudderth where the urn composition evolves as
follows: let $\pi:\left[0,1\right]\rightarrow\left[0,1\right]$, and denote by
$x_{n}$ the fraction of black balls after step $n$, then at step $n+1$ a black
ball is added with probability $\pi\left(x_{n}\right)$ and a white ball is
added with probability $1-\pi\left(x_{n}\right)$. Originally introduced to
mimic attachment under imperfect information, this model has found applications
in many fields, ranging from Market Share modeling to polymer physics and
biology. In this work we discuss large deviations for a wide class of
continuous urn functions $\pi$. In particular, we prove that this process
satisfies a Sample-Path Large Deviations principle, also providing a
variational representation for the rate function. Then, we derive a variational
representation for the limit
$\phi\left(s\right)=\lim_{n\rightarrow\infty}{\textstyle
\frac{1}{n}}\log\mathbb{P}\left(\left\{ nx_{n}=\left\lfloor sn\right\rfloor
\right\} \right),\, s\in\left[0,1\right]$, where $nx_{n}$ is the number of
black balls at time $n$, and use it to give some insight on the shape of
$\phi\left(s\right)$. Under suitable assumptions on $\pi$ we are able to
identify the optimal trajectory. We also find a non-linear Cauchy problem for
the Cumulant Generating Function and provide an explicit analysis for some
selected examples. In particular we discuss the linear case, which embeds the
Bagchi-Pal Model, giving the exact implicit expression for $\phi$ in terms of
the Cumulant Generating Function.

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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th