Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Conditioned local limit theorems for random walks defined on finite
  Markov chains   Let $(X_n)_{n\geq 0}$ be a Markov chain with values in a finite state space
$\mathbb X$ starting at $X_0=x \in \mathbb X$ and let $f$ be a real function
defined on $\mathbb X$. Set $S_n=\sum_{k=1}^{n} f(X_k)$, $n\geqslant 1$. For
any $y \in \mathbb R$ denote by $\tau_y$ the first time when $y+S_n$ becomes
non-positive. We study the asymptotic behaviour of the probability $\mathbb P_x
\left( y+S_{n} \in [z,z+a] \,,\, \tau_y > n \right)$ as $n\to+\infty.$ We first
establish for this probability a conditional version of the local limit theorem
of Stone. Then we find for it an asymptotic equivalent of order $n^{3/2}$ and
give a generalization which is useful in applications. We also describe the
asymptotic behaviour of the probability $\mathbb P_x \left( \tau_y = n \right)$
as $n\to+\infty$.

22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol