Strong invariance principle for a counterbalanced random walk

We study a counterbalanced random walk \(\check{S}_{n}=\check{X}_{1}+\cdots+\check{X}_{n}\) $${\check{S}}_n={\check{X}}_1+\cdots +{\check{X}}_n$$ , which is a discrete time non-Markovian process and \(\check{X}_{n}\) $${\check{X}}_n$$ are given recursively as follows. For n ≥ 2, \(\check{X}_{n}\) $${\check{X}}_n$$ is a new independent sample from some fixed law μ ≠ 0 with a fixed probability p, and \(\check{X}_{n}=-\check{X}_{v(n)}\) $${\check{X}}_n=-{\check{X}}_{v(n)}$$ with probability 1 − p, where v(n) is a uniform random variable on {1, ⋯, n − 1}. We apply martingale method to obtain a strong invariance principle for Š_n.