Random walk following $N(\mu,\sigma^2)$

Question

Consider a general random walk

$$ X_n=x_0+\sum_{k=1}^{n} J_k $$

where the jumps $J_k$ are IID RV's following $N(\mu,\sigma^2)$. Find the distribution of $X_n$ and:

$$\mathbb{P}(X_n \leq x, X_m \leq y) $$

where $n<m$. I don't know how to start, I'm just learning what random walks are.

Answer 1

The key lemma we need is:

Lemma. If $X\sim\mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim\mathcal N(\mu_2,\sigma_2^2)$ are independent, then $$X+Y\sim\mathcal N(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2).$$

The proof is easy -- just look at the moment generating functions.

This lemma immediately tells us that $X_n\sim\mathcal N(x_0+n\mu,n\sigma^2)$.

For the other question, by the Markov property $$f_{X_n,X_m}(x,y)=f_{X_m\mid X_n}(y\mid x)\cdot f_{X_n}(x).$$ Of course, we know the density of $X_n$. But conditional on $X_n=x$, the Markov property implies that $X_m\sim\mathcal N(x+(m-n)\mu,(m-n)\sigma^2)$. So we can find the joint density of $X_n$, $X_m$, and integrating over the appropriate region will give you $\mathbb P(X_n\leq x,X_m\leq y)$.