15.1.2 Theory of random processes

Problem: Let \( S_{0}=0, S_{k}=\xi_{1}+\cdots+\xi_{k}, 1 \leq k \leq n \), where \( \xi_{1}, \ldots, \xi_{k} \) are independent normally distributed, \( \mathcal{N}(0,1) \), random variables. Let \( \phi(x)=P\left\{\xi_{1} \leq x\right\} \), \( \mathcal{F}_{k}=\sigma\left(\xi_{1}, \ldots, \xi_{k}\right), 1 \leq k \leq n, \mathcal{F}_{0}=\{\emptyset, \Omega\} \). Show that for any \( a \in R \) the sequence \( X=\left(X_{k}, \mathcal{F}_{k}\right)_{0 \leq k \leq n} \) with \( X_{k}=\phi\left(\frac{a-S_{k}}{\sqrt{n-k}}\right) \quad \) is a martingale.