Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hitting probabilities of constrained random walks representing tandem
  networks   Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having
increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with
probabilities $\lambda$, $\mu_1$, $\mu_2$,...,$\mu_d$, where
$\{e_1,e_2,..,e_d\}$ are the standard basis vectors. The process $X$ is assumed
stable, i.e., $\lambda < \mu_i$ for all $i=1,2,3,...,d.$ Let $\tau_n$ be the
first time the sum of the components of $X$ equals $n$. We derive approximation
formulas for the probability ${\mathbb P}_x(\tau_n < \tau_0)$. For $x \in
\bigcup_{i=1}^d \Big\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j)$ $> \left(1 -
\frac{\log \lambda/\min \mu_i}{\log \lambda/\mu_i}\right) \Big\}$ and a
sequence of initial points $x_n/n \rightarrow x$ we show that the relative
error of the approximation decays exponentially in $n$. The approximation
formula is of the form ${\mathbb P}_y(\tau < \infty)$ where $\tau$ is the first
time the sum of the components of a limit process $Y$ is $0$; $Y$ is the
process $X$ as observed from a point on the exit boundary except that it is
unconstrained in its first component (in particular $Y$ is an unstable
process); $Y$ and ${\mathbb P}_y(\tau< \infty)$ arise naturally as the limit of
an affine transformation of $X$ and the probability ${\mathbb P}_x(\tau_n <
\tau_0).$ The analysis of the relative error is based on a new construction of
supermartingales. We derive an explicit formula for ${\mathbb P}_y(\tau <
\infty)$ in terms of the ratios $\lambda/\mu_i$ which is based on the concepts
of harmonic systems and their solutions and conjugate points on a
characteristic surface associated with the process $Y$; the derivation of the
formula assumes $\mu_i \neq \mu_j$ for $i\neq j.$

11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th