"Independent observations" via measure theory

Question

I'm reading Chernoff's paper "A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations," and am trying to understand it in terms of measure theory. On page 495, it says:

"$S_n$ is the sum of $n$ independent observations $X_1,X_2,\ldots,X_n$ on a chance variable $X$."

Which of the following is the correct interpretation?

1. Let $M$ be a measurable space and let $X:M\to\mathbb{R}$ be a measurable function. Then $S_n:M\times\cdots\times M\to\mathbb{R}$ is the function $S_n(p_1,\ldots,p_n)=X(p_1)+\cdots+X(p_n)$
2. Let $(M,\mu)$ be a probability space and let $X:M\to\mathbb{R}$ be a measurable function. Let $X_1,\ldots,X_n$ be real-valued measurable functions on $M$ such that $(X_i)_\sharp\mu=X_\sharp\mu$ as measures on $\mathbb{R}$ for each $i$ and such that $(X_i,X_j)_\sharp\mu=(X_i)_\sharp\mu\times(X_j)_\sharp\mu$ as measures on $\mathbb{R}^2$ for each $i,j$. Define $S_n:M\to\mathbb{R}$ by $S_n(p)=X_1(p)+\cdots+X_n(p).$
3. Something else?

Answer 1

Since Chernoff (1952) already uses the letter $M$ to denote the moment generating function, it is preferable to use a different symbol for the underlying sample space. Thus, I will suppose that the problem is located in the probability space $(\mathcal{S}, \mathscr{S}, P)$. When dealing with random variables that are generated as functions of other random variables, it is useful to start by looking at all of them in the same overarching probability space. If it is helpful, you can then look at marginal measures within this space. When looking at the random variables in the overarching space, here are a few points to keep in mind.

---

1. If you have some set of real random variables $X_1,...,X_n$, then each of these random variables is a (measureable) mapping $X_i: \mathcal{S} \rightarrow \mathbb{R}$ from the sample space to the real numbers. A single outcome $\omega \in \mathcal{S}$ in the sample space determines the value of all of these random variables.

2. Now suppose you a some real random variable $S_n \equiv g(X_1,...,X_n)$ defined by a (measureable) function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ on underlying real random variables $X_1,...,X_n$. This random variable is a composite function, which reduces down to $S_n: \mathcal{S} \rightarrow \mathbb{R}$, and is given by: $$S_n(\omega) = g(X_1(\omega), .... , X_n(\omega)) \quad \quad \quad \text{for all } \omega \in \mathcal{S}.$$ There is a single outcome $\omega$ that determines all the random variables in the problem. Each random variable in the problem, including $S_n$, is a mapping from the sample space to the real numbers. Note also that this representation holds regardless of whether or not the underlying random variables are independent, and regardless of the function $g$.

3. The distribution of $S_n$ is determined by the underlying probability measure $P$ on the class of events $\mathscr{S}$. The cumulative distribution function for this random variable can be written as:$$F_{S_n}(s) = P(\mathcal{E}_s) \quad \quad \quad \mathcal{E}_s \equiv \{ \omega \in \mathcal{S} | g(X_1(\omega), .... , X_n(\omega)) \leqslant s \}.$$ The marginal probability measure for $S_n$ is the probability measure induced by this distribution function. Again, this result holds regardless of whether or not the underlying random variables are independent, and regardless of the function $g$.

4. If the random variables $X_1,...,X_n$ are mutually independent, it is useful to frame the analysis in terms of their marginal behaviour. To do this, let $(\mathbb{R}, \mathscr{R}, \mu_i)$ denote the marginal probability space for a single one of these underlying random variables $X_i$. We can write the cumulative distribution function for a single random variable as $F_{X_i}(x_i) = \mu_i((-\infty,x_i])$. Since the random variables are independent, the cumulative distribution function for the vector of underlying random variables can now be written as:$$F_{\mathbf{X}_n}(\mathbf{x}_n) = \prod_{i=1}^n \mu_i((-\infty,x_i]).$$ The probability measure for this vector of values is the measure induced by this distribution function. This appears to be the same thing you are trying to get across in your second interpretation, though I confess that I could not follow your notation.