Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\left(\xi_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically Bernoulli \((p, 1-p)\)-distributed random variables with values \(\pm 1\), i.e. such that \(\mathbb{P}\left(\xi_{n}=1ight)=p\) and \(\mathbb{P}\left(\xi_{n}=-1ight)=1-p-\) this can be constructed as in Scholium 23.4 . Set \(S_{n}:=\xi_{1}+\cdots+\xi_{n}\). Then \(((1-p) / p)^{S_{n}}\) is a martingale w.r.t. the filtration given by \(\mathscr{A}_{n}:=\sigma\left(\xi_{1}, \ldots, \xi_{n}ight)\).