An overview of Brownian Motion in Probability Theory

I will try providing a qualitative overview of Brownian motion as it may be taught in a post-graduate measure theoretic probability class. I will exactly follow Morters and Peres. Additional notes may come from Karatzas if I have the time.

Definition to provide context

BM is infinitesimally random normally incremented motion. Denoted $(B_t)_{t \ge 0}$, has 3 defining tenets:

1. $B_0 \in \RR$ (deterministic starting point)
2. $B_t - B_s \sim \N(0,t-s)$ independently of $B_s$ $\forall t,s$ (independent normal increments)
3. function $t \to B_t$ is continuous (continuity of almost all paths)

The 1st tenet, really, follows from the 2nd because of the stability of $\N$ family.

Now if a particle moves at every instant, does it even move, or does it remain stagnant due to "chronic indecision" (every attempted move is instantly discarded by the whim of the immediate next moment)! Surprisingly, math deems it legal to create a stochastic process whose no matter how small a timespan you assess, you shall always find a random normal increment independent of the past. And it says that the process does move. We will discuss why.

2 levels of studying BM

1. Calculus aspect
2. Measure theoretic aspect

These levels are indeed interconnected. For example:

Defining stop time requires measurability notions. But once the definition is well understood, many applications of stop time and its related theorems flow without explicit reference to measurability conditions anymore. Mostly because "is this object measurable with respect to this field" is answered implicitly, heuristically. We will see how.

Measure theory portions of BM discussion can be significantly more challenging because they will often involve lots of real analytic probability (an actual domain; see Real Analysis and Probability by Richard Dudley).

To prepare you (and myself) honestly for these looming, risky parts, I will mention explicitly, how much you need to be scared before you begin every section. This is same as the percentage of measure theory there is.

1. Definition and first properties

80% measure

Constructive definition (Levy's construction) of BM proves that BM actually "exists" (mathematically legal). Levy says the following:

1. We need to define BM over an entire interval, say $[t_1, t_2]$. But work with $[0,1]$ and translate accordingly later.
2. Take a finite set of timepoints from an entire continuous interval over which we

Path properties: 80% measure

Berry-Esseen Theorem

Personally, I've always liked this name: Berry-Esseen. So sleek, so stylish. A certain Blackberry vibe to it. The theorem, however, owes its name to Andrew C. Berry and Carl-Gustav Esseen, both of whom independently proved (the first iteration of) the undermentioned result in the 1940s.

What is the Berry-Esseen Theorem about? It is a theorem that declares a (1) uniform bound on the extent of (2) "normality" of the sample mean (3) under CLT under the (4) existence of the 3rd order absolute moment. What do I mean? Let's go step by step:

Theorem: Berry-Essen Theorem (BET)

Setup: RVs $X_1, X_2, \dots$ i.i.d. have $\E X_i =0$, $\mathbb E X_i^2 = \sigma^2 >0$ and $\mathbb E |X_i|3 = \gamma < \infty$.

Notation: $\bar X_n := \frac{X_1 + \dots + X_n}{n}$. Let $F_n$ be the CDF of $\frac{\bar X_n}{\sigma/\sqrt n}$.

Existing: From CLT we already know $\frac{\bar X_n}{\sigma/\sqrt{n}} \to \mathcal N (0,1)$, i.e. $F_n \to \Phi$.

BET further states:

\[\sup_{x \in \RR} |F_n(x) - \Phi(x)| \leq \frac{C\gamma}{\sigma^3 \sqrt n}$\]

Explainer:

1. The larger a sample, the more the "normality" in the distribution of its sample mean.
2. BET is a bound on how much the distribution of $\sqrt n \bar X_n/ \sigma$ (standardized sample mean) can deviate from normality.
3. The bound is supremal (on the worst possible deviation), and uniform (the bound works for $x$).

Note:

1. Esseen (1956) proved that $C \geq \frac{\sqrt{10} + 3}{6 \sqrt{2 \pi}}$. Recent expositions have proved $C \leq 0.4748$. But nobody knows what $C$ exactly is.
2. $F_n$ is the (small sample) CDF $\sqrt n \bar X/ \sigma$ based on $X_1, \dots, X_n$. It is not an empirical CDF, because of technically $F_n(t) = \E \left[\sum_{i=1}^n \mathbb I(X_i \le t)\right]$ is a deterministic function: the $\E$ makes it non-random.

Proof Sketch

To summarize it,

After BET