  In this paper we introduce randomized branching as a tool for parameterized
approximation and develop the mathematical machinery for its analysis. Our
algorithms improve the best known running times of parameterized approximation
algorithms for Vertex Cover and $3$-Hitting Set for a wide range of
approximation ratios. One notable example is a simple parameterized random
$1.5$-approximation algorithm for Vertex Cover, whose running time of
$O^*(1.01657^k)$ substantially improves the best known runnning time of
$O^*(1.0883^k)$ [Brankovic and Fernau, 2013]. For $3$-Hitting Set we present a
parameterized random $2$-approximation algorithm with running time of
$O^*(1.0659^k)$, improving the best known $O^*(1.29^k)$ algorithm of [Brankovic
and Fernau, 2012].
  The running times of our algorithms are derived from an asymptotic analysis
of a wide class of two-variable recurrence relations of the form: $$p(b,k) =
\min_{1\leq j \leq N} \sum_{i=1}^{r_j} \bar{\gamma}_i^j \cdot p(b-\bar{b}^j_i,
k-\bar{k}_i^j),$$ where $\bar{b}^j$ and $\bar{k}^j$ are vectors of natural
numbers, and $\bar{\gamma}^j$ is a probability distribution over $r_j$
elements, for $1\leq j \leq N$. Our main theorem asserts that for any
$\alpha>0$, $$\lim_{k \rightarrow \infty } \frac{1}{k} \log p(\alpha k,k) =
-\max_{1\leq j \leq N} M_j,$$
  where $M_j$ depends only on $\alpha$, $\bar{\gamma}^j$, $\bar{b}^j$ and
$\bar{k}^j$, and can be efficiently calculated by solving a simple numerical
optimization problem. To this end, we show an equivalence between the
recurrence and a stochastic process. We analyze this process using the Method
of Types, by introducing an adaptation of Sanov's theorem to our setting. We
believe our novel analysis of recurrence relations which is of independent
interest is a main contribution of this paper.
