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Let $\{X_i\}_{i \in \mathbb{N}}$ be a sequence of i.i.d random variables taking value in the set $\{-1,1\}$ with $P(X_i = 1) = \frac{1}{2}$. Let $S_n = \sum\limits_{i=1}^n X_i$, and for $k \in \mathbb{N}$ define the random process $B^k$ as $$ B^k(t) = \frac{S_{[kt]}}{\sqrt{k}} \qquad t \in \mathbb{R} $$

where $[x]$ is the integer part of $x$.

I know that $(B^k(t))_{t \in \mathbb{R}}$ converges to the standard Brownian motion in distribution, does it also converges pointwise? i.e. is there a random process $\{B(t)\}_{t\in \mathbb{R}}$ so that $$ \forall t \in \mathbb{R} \quad \lim_{k \to \infty} B^k(t) = B(t) \quad \text{a.s} $$ and further that $B(t)$ is the Brownian motion.

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  • $\begingroup$ If the pointwise limit exists, then it must be BM. So the question really is if the limit exists. $\endgroup$
    – Miheer
    May 5, 2017 at 15:10
  • $\begingroup$ A related question seems to be : is there pointwise convergence in the central limit theorem $\endgroup$
    – Miheer
    May 5, 2017 at 15:19

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No, it doesn't. By the Kolmogorov 0-1 law, if the a.s. (or even i.p.) limit exists, it is a constant, which is absurd.

To see this, let's take $t=1$ for simplicity. Suppose that $S_k / \sqrt{k} \to Y$ a.s. Fix $j$ and write, for $k \ge j$,

$$\frac{S_k}{\sqrt{k}} = \frac{X_1 + \dots + X_{j-1}}{\sqrt{k}} + \frac{X_{j} + \dots + X_k}{\sqrt{k}}$$

As $k \to \infty$, the first term goes to zero a.s. (since the numerator does not depend on $k$). So we have

$$Y = \lim_{k \to \infty} \frac{X_{j} + \dots + X_k}{\sqrt{k}}$$

which shows that $Y \in \sigma(X_j, X_{j+1}, \dots)$. But $j$ was arbitrary, so this shows that $Y$ is a tail random variable. Kolmogorov says the tail $\sigma$-field is almost trivial, meaning that $Y$ is a.s. equal to a constant.

Alternatively, the law of the iterated logarithm for random walk will also show you that, almost surely, $\limsup S_k / \sqrt{k} = +\infty$ and $\liminf S_k / \sqrt{k} = -\infty$.

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    $\begingroup$ What is fun is that Steven Shreve explicitly pretends having constructed the Brownian motion as the limit of these scaled symmetric random walks in Stochastic Calculus for Finance II in 3.1.1. Page 93. $\endgroup$
    – Olórin
    Nov 13, 2019 at 22:51
  • $\begingroup$ Can I quote this result in a paper? Should I reference it as a post on MSE? $\endgroup$ Mar 17, 2023 at 9:57
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    $\begingroup$ @JanStuller: You certainly can if you like; there's a cite button under the post to generate a bibtex entry. But I feel like it is pretty well known that there cannot be pointwise convergence in the CLT - as in my last line, that's in some ways the main point of the LIL. $\endgroup$ Mar 17, 2023 at 15:19
  • $\begingroup$ @Olórin: To be fair to Shreve, he just says "We obtain Brownian motion as the limit of the scaled random walks $W^{(n)}(t)$ of (3.2.7) as $n \to \infty$". He doesn't say "pointwise limit" and he presumably means "limit in distribution", which is quite correct. (Anyway, I think it's more of an offhand comment than a mathematical claim: the convergence in distribution of the process, i.e. Donsker's theorem or functional CLT, is not formally stated or proved in the book.) $\endgroup$ Mar 17, 2023 at 15:29
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You cannot do this but if you allow constructing $B^k$ with different copies of i.i.d. random variables, then by Skorokhod's representation theorem you can find a space such that the point-wise convergence hold (for different $B^k$ the $X_i$ are different).

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  • $\begingroup$ see Theorem 13.8 in Breiman's book "Probability" for a nice self-contained presentation. $\endgroup$
    – am70
    Jun 12, 2022 at 18:32

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