$\int_0^1 X_i di$, where $X_i \sim \log N(\mu,\sigma^2)$

Question

Intuition tells me this question has something to do with stochastic process. Since I'm not quite familiar with it, I might misuse some terms or definitions.

Supposed that $X_i$, $i \in [0,1]$ are i.i.d random variables, where $X_i \sim \log N(\mu, \sigma^2)$.
Then, is $$ \int_0^1 X_i di$$ well defined?

If yes, then is it a random variable as well, or is it some sort of "expectation" ?

Thanks in advance!

Answer 1

This is an important question related to the law of large numbers for continuum of variables. In the context of finitely many i.i.d. random variables we have assertions of the type

$$\sum_{i=1}^N X_i \rightarrow \mu \text{ as } N\rightarrow\infty$$

under suitable conditions, where $\mathbb{E}[X_i]=\mu$. The convergence is almost sure (strong law of large numbers) or in probability (weak law of large numbers).

The question you are asking is then whether some sort of law of large numbers for i.i.d. random variables exists for continuum of random variables. If we could invoke the law of large number then we would asserts that $$\int_0^1 X_i di \quad ``=" \quad\mu, \qquad \qquad(*)$$ where $\mathbb{E}[X_i]=\mu$. I put the equality sign in quotation marks since LHS is random while the RHS is constant so we need to ask ourselves in what sense we want this equality holds.

---

Definitions

To cast this problem formally, we need to define $X_i(\omega)$ as a function on appropriate measure space. We need index space $([0,1],\mathcal{I},\lambda)$, where $\lambda$ is a Lebesgue measure, and a sample space $(\Omega,\mathcal{F},P)$. Then, we define a product probability space in the usual way as $(I\times\Omega,\mathcal{I}\otimes\mathcal{F},\lambda\otimes P)$.

Consider a function $X(i,\omega)$ defined on $I\times\Omega$. Let $X_{\omega}$ represent a function $X(\cdot,\omega)$ which is a function on $[0,1]$ and let $X_{i}$ represent a function $X(i,\cdot)$ which is a function on $\Omega$. In addition, assume that $X(i,\cdot)$ are essentially pairwise independent. That is, for $\lambda$-almost all i, $X_i$ is independent of $X_j$ for $\lambda$-almost all $j\in [0,1]$.

Almost sure equality fails

Suppose that we want $(*)$ to hold almost surely for $X(i,\omega)$ defined above. Then we quickly run into two problems describe by Judd 1985.

1. The function $X_{\omega}(\cdot)$ is not measurable on $[0,1]$ for $\omega\in N$, where $N$ itself is non-measurable.
2. Even if that problem is fixed by suitable extension of underlying measure space, so $X_{\omega}(\cdot)$ is measurable, $(*)$ will not hold in general.

Below, I state two results that point to the problem with defining $(*)$ in almost sure sense. The first result is due to Sun, 2006.

Theorem (Sun, 2006) Suppose that $X$ is a function on $I\times\Omega$ such that $X$ is measurable on $(I\times\Omega,\mathcal{I}\otimes\mathcal{F},\lambda\otimes P)$. Furthermore, suppose that $X_i$'s are essentially pairwise independent. Then, for $\lambda$-almost all $i$, $X_i$ is a constant random variable.

The second result is due to Uhlig(1996).

Theorem (Uhlig, 1996) Suppose that $(X(i),i\in[0,1])$ is a collection of identical, pairwise independent r.v. with mean $\mu$. Suppose that for some $\omega$ it is true that $X(\cdot,\omega)$ is measurable on $[0,1]$ and that for all $a,b$ such that $a,b\in[0,1]$ and $a\leq b$ we have $$\int_a^b X_i(\omega) di=\mu(b-a)\text{ a.s.}$$ Then $$X(i,\omega)=\mu$$ for almost all i.

In other words, if a law of large numbers is supposed to hold for a continuum of i.i.d. random variables in almost sure sense, then it has to be the case that $X_i(\omega)$ must be essentially constant random variables.

Solutions

There are two ways to get out of this problem.

1. One is to consider a weak law of large numbers and interpret (*) as meaning that the LHS converges to RHS in probability rather than almost surely. This is the approach taken by Uhlig(1996).
2. The other way was suggested by Sun, 2006 who shows that one can obtain strong law of large numbers but this has to be achieved by using non-standard analysis (nonstandard analysis).

References

1. Judd, "The law of large numbers with a continuum of IID random variables", Journal of Economic Theory, Volume 35, Issue 1, February 1985, Pages 19-25

2. Sun, "The exact law of large numbers via Fubini extension and characterization of insurable risks", Journal of Economic Theory, Volume 126, Issue 1, January 2006, Pages 31-69

3. Uhlig, "A law of large numbers for large economies," Economic Theory, February 1996, Volume 8, Issue 1, pp 41–50

Yeneng Sun has several papers on the this topic besides the once cited above.