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I would like to understand the following sentence from Baxter and Rennie's book Financial Calculus:

"There is a formal unity underlying the family - all the marginal distributions tend towards the same underlying normal structure"

It refers to the family $\{W_n(t), n \in \mathbb{N}\}$ of random walks. I believe it means that in the limit as $n \to \infty$, $W_n(t) \to W_t$, which is distributed as $N(0,t)$.

$W_n(t)=\frac{\sum_{i=1}^{nt} X_i}{\sqrt{n}}$ where each $X_i$ is a random variable taking values $+1$ or $-1$ with equal probabilities. My question is, how is $W_n(t)$ a marginal distribution? If it is a marginal distribution, what is the corresponding joint distribution?

Kristina
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1 Answers1

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If you have an $n$-dimensional random variable $(X_1, \ldots, X_n)$, then the distribution of one component $X_i$ is called a marginal distribution of the joint distribution of $(X_1, \ldots, X_n)$.

A random walk $(W_n(t))_{t \ge 0}$ (here $n$ is fixed) is the same thing, but instead of having finitely or countably many components, there is a "component" for each real number $t \ge 0$. So the distribution of $W_n(t)$ (for a single $t$) is a marginal distribution of the joint distribution of the whole walk $(W_n(t))_{t \ge 0})$ (over all $t$).

angryavian
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