# About a factor 2 in Donsker's CLT normalization

I have a question about the following paper:

Donsker, Monroe D. "An invariance principle for certain probability limit theorems." Mem. Amer. Math. Soc 6.1951 (1951): 12.

There is a factor of two throughout the paper that I don't understand. I'll state the question more precisely after some definitions. Thanks in advance for any assistance! I am probably just being stupid.

## Definitions

First, some definitions, from the paper.

• "We consider a sequence $S_1, S_2, S_3,...$ of partial sums of independent, identically distributed random variables $u_1,u_2,u_3,...$ each having mean $0$ and standard deviation $1$."

• "By Wiener space we mean here the space $C$ consisting of all continuous functions $x(t)$ defined on $0\le t\le 1$ $(x(0)=0)$ with Wiener measure imposed on $C$. $^{(2)}$ ..."

• "...$^{(2)}$ For the definition of Wiener measure used here see N. Wiener, Generalized Harmonic Analysis. Acta Math., vol. 55(1930), pp. 117-158, especially pages 214-234."

• "Let $k$ be a fixed positive integer, $\alpha$ and $\beta$ be vectors $\alpha:(\alpha_1,\alpha_2,...,\alpha_k)$ and $\beta:(\beta_1,\beta_2,...,\beta_k)$"

• "Let $\mathfrak{V}$ be any positive integer"

• "$n_{j,p} = \left[ \frac{ (j-1) n}{k} + \frac{pn}{k \mathfrak{V}} \right]$, for $j=1,2,...,k$ and $p=0,1,...,\mathfrak{V}$."

• "Let $F_n$ be the subset of $(u_1,u_2,...,u_n)$ such that for all $j=1,2,...,k$ \begin{align*} (2n)^{1/2} \alpha_j \le s_{n_{j,p}} \le (2n)^{1/2} \beta_j\ \ \ \ \ \ \ \ \ \ \ \ \ \ (p=0,1,...,\mathfrak{V})" \end{align*}

• "Let $D_{\mathfrak{V}}$ be the subset of $C$ such that for all $j=1,2,...,k$ \begin{align*} \alpha_j \le x\left(\frac{(j-1)\mathfrak{V}+p}{k\mathfrak{V}}\right) \le \beta_j\ \ \ \ \ \ \ \ \ \ \ \ \ (p=1,2,...,\mathfrak{V})" \end{align*}

## Statement

Using these definitions, he states the following:

"It is a consequence of the multidimensional central limit theorem that ... \begin{align*} \underset{n\rightarrow\infty}{\lim} P\{F_n\} = P\{D_\mathfrak{V}\}." \end{align*}

## Question

So here is my question: Is this statement correct given the definitions above? In particular, should the factors of "$(2n)^{1/2}$" in the definition of $D_{\mathfrak{V}}$ be replaced with "$(n)^{1/2}$?" In my present, unenlightened state, one of the following two options seems likely:

1. The answer is yes, and the factor of two is misplaced.
2. The definition of Wiener measure used in this paper is unconventional (I don't have access to Generalized Harmonic Analysis by Norbert Wiener, from which Donsker gets the definition used in this paper).

## P.S.

This is my first post, so pardon my clunky style and possible violations of any rules or customs. I'm happy to make any changes recommended by the community.

If anyone else has direct access to this paper, I have another question.

NW probably defines the standard normal density as being proportional to $x\mapsto\mathrm e^{-x^2}$ (like many analysts do), not to $x\mapsto\mathrm e^{-x^2/2}$ (the choice of many probabilists), in which case the extraneous factor $\sqrt2$ is needed.