Repeated-observation sampling models with IID and finite-variance hypotheses

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space (Probability measures and probability spaces) and let $n\ge 1$ be a natural number. A repeated-observation sampling model for a real quantity consists of random variables $Y_0,\dots ,Y_{n-1}:\mathrm{\Omega }⟶\mathbb{R}$ (Random elements and real random variables) together with the following model hypotheses, each of which is adopted as part of the model and is not a consequence of the data:

1. (IID hypothesis) the family $(Y_k{)}_{k<n}$ is independent and identically distributed in the sense of Identical distribution and IID families; that is, the whole family is mutually independent and the laws of the $Y_k$ all agree;
2. (second-moment hypothesis) $E\left[Y_0^2\right]<+\mathrm{\infty }$.

Under hypotheses 1 and 2 the numbers $\mu :=E[Y_0]\in \mathbb{R},{\sigma }^2:=\mathrm{Var}(Y_0)\in [0,+\mathrm{\infty })$ are finite by Moments, variance, and covariance on a probability space, and hypothesis 1 makes them the common mean and variance of the family; in particular $E[Y_k]=\mu$ and $\mathrm{Var}(Y_k)={\sigma }^2$ for every $k<n$. The sample mean of the model is the random variable $\bar{Y}:=\frac{1}{n}{\sum }_{k=0}^{n-1}{Y}_k\mathrm{.}$

Conventions and scope

The data of the model are the probability space, the variables $Y_k$, and the adopted hypotheses; hypotheses 1 and 2 can fail for the same family of variables on the same probability space, so they must be stated, not inferred from the notation. The definition assumes no particular law for the $Y_k$: it neither assumes a normal law, nor a finite fourth moment, nor a continuous distribution. Hypotheses of that kind, and any conclusion needing them, must be stated separately at their point of use.

A sampling model is a mathematical description of repeated observation. It does not assert that any physical sequence of measurements satisfies the IID and second-moment hypotheses, and it attaches no value to $\mu$ or ${\sigma }^2$ beyond the one supplied by the model.