arXiv subjects

Shengli Liang

Publications and source records attributed to Shengli Liang.

3 recordsLinked to original sources

arXiv2023

Conditional central limit theorem for critical branching random walk

Consider a critical branching random walk on $\mathbb{R}$. Let $Z^{(n)}(A)$ be the number of individuals in the $n$-th generation located in $A\in \mathcal{B}(\mathbb{R})$ and $Z_{n}:=Z^{(n)}(\mathbb{R})$ denote the population of the $n$-th generation. We prove that, under some conditions, for all $x\in \mathbb{R}$, as $n\to \infty$, $$\mathcal{L}\left(\frac{Z^{(n)}(-\infty, \sqrt{n} x]}{n} ~\bigg |~ Z_{n}>0\right) \Longrightarrow\mathcal{L}\left(Y(x)\right),$$ where $\Rightarrow$ means weak convergence and $Y(x)$ is a random variable whose distribution is specified by its moments.

math.PR↗

arXiv2023

Convergence of the derivative martingale for the branching random walk in time-inhomogeneous random environment

Consider a branching random walk on the real line with a random environment in time (BRWRE). A necessary and sufficient condition for the non-triviality of the limit of the derivative martingale is formulated. To this end, we investigate the random walk in time-inhomogeneous random environment (RWRE), which related the BRWRE by the many-to-one formula. The key step is to figure out Tanaka's decomposition for the RWRE conditioned to stay non-negative (or above a line), which is interesting itself as well.

math.PR↗

arXiv2022

Random walks in time-inhomogeneous random environment conditioned to stay positive

We consider a random walk $\{S_n\}_{n\in \mathbb{N}}$ in time-inhomogeneous random environment $ξ$. For almost each realization of $ξ$, we formulate a quenched harmonic function, based on which we can define the random walk in random environment conditioned to stay positive by the Doob's $h$-transform. Furthermore, we prove a quenched invariance principle for the conditioned random walk for almost each realization of $ξ$.

math.PR↗