Consider a symmetric simple random walk $\left(Z_{n}\right)_{n \in \mathbb{Z}^{+}}$taking values in statespace $I=h \mathbb{Z}^{2} \equiv\{(i h, j h): i, j \in \mathbb{Z}\}$, where $h \equiv N^{-1}(N$ an integer $)$. Writing $Z_{n} \equiv\left(X_{n}, Y_{n}\right)$, the transition probabilities are given by

$$P\left(\Delta Z_{n}=(h, 0)\right)=P\left(\Delta Z_{n}=(0, h)\right)=P\left(\Delta Z_{n}=(-h, 0)\right)=P\left(\Delta Z_{n}=(0,-h)\right)=\frac{1}{4},$$

where $\Delta Z_{n} \equiv Z_{n}-Z_{n-1}$.

What does it mean to say that $\left(M_{n}, \mathcal{F}_{n}\right)_{n \in \mathbb{Z}^{+}}$is a martingale? Find a condition on $\theta$ and $\lambda$ such that

$$M_{n}=\exp \left(\theta X_{n}-\lambda Y_{n}\right)$$

is a martingale. If $\theta=i \alpha$ for some real $\alpha$, show that $M$ is a martingale if

$$e^{-\lambda h}=2-\cos (\alpha h)-\sqrt{(2-\cos (\alpha h))^{2}-1}$$

Suppose that the random walk $Z$ starts at position $(0,1) \equiv(0, N h)$ at time 0 , and suppose that

$$\tau=\inf \left\{n: Y_{n}=0\right\} .$$

Stating fully any results to which you appeal, prove that

$$E \exp \left(i \alpha X_{\tau}\right)=e^{-\lambda}$$

where $\lambda$ is as given at $(*)$. Deduce that as $N \rightarrow \infty$

$$E \exp \left(i \alpha X_{\tau}\right) \rightarrow e^{-|\alpha|}$$

and comment briefly on this result.