We study the second order of the number of excursions of a simple random walk with a bias that drives a return toward the origin along the axes introduced by P. Andreoletti and P. Debs [2]. This is a crucial step toward deriving the asymptotic behavior of these walks, whose limit is explicit and reveals various characteristics of the process: the invariant probability measure of the extracted coordinates away from the axes, the one stable distribution arising from the tail distribution of entry times on the axes and, finally, the presence of a Bessel process of dimension 3, which implies that the trajectory can be interpreted as a random path conditioned to stay within a single quadrant.
