Comparison of limit theorems

Question

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and on which we define a Brownian motion $(\mathbb{B}(t))_{t \geq 0}$. Further let $(B_n)_{n \in \mathbb{N}}$ be a sequence, which satisfies the properties of a Brownian motion for integer values (or in other words a Brownian motion for integer values). I guess an example would be the partial sum of process of a sequence of iid standard normal distributed random variables.

Let now $(X_n)_{n \in \mathbb{N}}$ be an iid sequence of random variables and let $S_n = \sum_{j=1}^n X_j$ denotes its partial sum process and let $p > 2$. Now compare the two limit theorems for $S_n$:

1. There exists a probability space $(\Omega', \mathcal{A}', \mathbb{P}')$, a sequence $(S_n')_{n \in \mathbb{N}}$ on this probability space such that $(S_n)_{n \in \mathbb{N}} \overset{\mathcal{D}}{=} (S_n')_{n \in \mathbb{N}}$ and a Brownian motion $\mathbb{B}(\cdot)'$ on that space such that$$\frac{S_n' - \mathbb{B}'(n)}{n^{1/p}} \to 0 \,\,a.s.$$

2. There exists a probability space $(\Omega', \mathcal{A}', \mathbb{P}')$, a sequence $(S_n')_{n \in \mathbb{N}}$ on this probability space such that $(S_n)_{n \in \mathbb{N}} \overset{\mathcal{D}}{=} (S_n')_{n \in \mathbb{N}}$ and a sequence $B_n'$ that is a Brownian motion for integer values on that space such that $$\frac{S_n' - B_n'}{n^{1/p}} \to 0 \,\,a.s.$$

Do these two limit theorems contain the same information? The former looks stronger (at least to my unexperienced eye), but somehow I think that they are equivalent. Could someone help me out?

Ah, and if someone is interested, the first limit theorem is known as the Komlos-Major-Tusnady approximation.

Answer 1

The two statements are equivalent. For their validity, you need to assume that $E(X_1)=0$ and $E|X_1|^p<\infty$. To show the equivalence it suffices to assume that condition (2.) holds and derive condition (1.)

So suppose condition (2.) holds. By enlarging the probability space if necessary, we may assume that infinitely many Brownian motions in the unit interval $(W_n)_{n=1}^\infty$ (independent of each other and the process $(S_n', B_n')_{n \ge 1} $) are defined on the probability space $\Omega'$ in condition (2.). Now define a Brownian motion $\mathbb{B}(\cdot)$ on this space by $$\forall t \in [0,1] \quad \mathbb{B}(n-1+t)=B_{n-1}'+W_n(t)+t(B_n'-B_{n-1}'-W_n(1)) \,.$$ In other words, we interpolate between the given values at integer times using independent Brownian bridges [1]. It is easy to check that $\mathbb{B}(\cdot)$ is indeed Brownian motion on $[0,\infty)$, so condition (1.) is satisfied.

[1] https://en.wikipedia.org/wiki/Brownian_bridge