# exact meaning of uniform integrability for empirical distributions

Suppose we have $$n$$ non-negative integer-valued random variables $$X_1,\ldots,X_n$$ and consider the empirical distribution $$Q := \frac{1}{n}\sum_{i=1}^n \delta_{X_i}.$$ We equip any probability mass function $$q \in \mathcal{P}(\mathbb{Z}_+)$$ with the usual $$\ell^1$$ norm $$\|q\|:= \sum_{k=0}^\infty q_k$$. I am confused on the precise meaning of the statement that the empirical distribution $$Q$$ is uniformly integrable (notice that $$Q$$ depends on $$n$$). In the typical setting, we say that a collection of random variables $$(X_n~\colon n \in \mathbb N)$$ is U.I. (Uniformly Integrable) if there exists some $$K >0$$ so that $$\sup_{n\in \mathbb N} \mathbb{E}(|X_n|\mathbf{1}_{|X_n| \geq K}) < \epsilon$$ for each pre-fixed $$\epsilon > 0$$. But how does this general definition translates into the aforementioned setting really bothers me... Thanks for any help!

• What is your reference? Anyway, $0\le Q_n\le 1$ and so $\{Q_n\}$ is trivially u.i. Oct 27 '20 at 7:47
• No, I disagree with the statement that $\{Q_n\}$ is trivially U.I. the reference I use is this paper arxiv.org/pdf/1804.04608.pdf Immediately below Proposition 3 (page 6), it is mentioned that "if Q(0) is U.I., then ...", so the U.I. is "not free" to obtain. Oct 27 '20 at 17:55
• en.wikipedia.org/wiki/Uniform_integrability#Related_corollaries Oct 27 '20 at 17:59
• I looked at wiki as well, but that doesn't really help me understand the case of empirical distribution here...What U.I. means exactly in this setting is not clear to me Oct 27 '20 at 18:01
• Thanks for your comments! It now makes more sense to me! Dec 21 '20 at 0:36

I'm going to reference arxiv.org/pdf/1804.04608.pdf, which was cited in the comments as the motivation for this question. The paper uses the following notation:

• $$\Omega = \{(\eta_i)_{i=1,\dots,n}\in \mathbb{Z}_+: \sum_{i=1}^n \eta_i = m\}$$. Because we vary $$m$$ and $$n$$, I'm going to instead write $$m_n$$ and $$\Omega^m_n$$. The paper seems to be implying that $$(\eta_i)_{i\in\mathbb{N}}$$ is some fixed sequence.
• $$Q(0) = \frac{1}{n}\sum_{i=1}^n \delta_{\eta_i}$$ is the empirical measure. Again, I will write $$Q_n = \frac{1}{n}\sum_{i=1}^n \delta_{\eta_i}$$ for clarity, omitting the $$(0)$$ as this question does not regard the dynamics of the process introduced in the paper. For each $$n$$, we regard $$Q_n$$ as an element of $$\ell^1(\mathbb{Z}_+)$$.
• $$Q_n \to q$$ in $$\ell^1(\mathbb{Z}_+)$$, $$\frac{m_n}{n} \to \rho$$ and $$\lambda = \sum_{k=0}^\infty kq_k$$.

We ignore the dynamics introduced in the paper. We can clearly see that each $$Q_n$$ is a probability measure (I'm assuming the configuration $$(\eta_i)_{i\in\mathbb{N}}$$ is fixed and non-random. If I misread that, then the argument only changes slightly). Furthermore, convergence in $$\ell^1$$ is equivalent to total variation convergence, so $$q$$ is also a probability measure. So, we can assign to each $$n$$ a random variable $$Y_n := \eta_{U_n}$$ where $$U_n$$ is sampled from $$\{1,\dots,n\}$$ uniformly at random. Then $$Y_n$$ has distribution $$Q_n$$. Let $$Y$$ be some random variable sampled from $$q$$. Then $$Y_n \to Y$$ in total variation. For each $$n$$, $$\mathbb{E}[Y_n] = \frac{1}{n}\sum_{i=1}^n \eta_i = \frac{m_n}{n} \to \rho$$, and $$\mathbb{E}[Y] = \sum_{k=1}^\infty kq_k = \lambda$$.

The paper states that $$\rho = \lambda$$ if $$Q(0)$$ is uniformly integrable. What they mean is that if the sequence of random variables $$\{Y_n\}_{n \in \mathbb{N}}$$ is uniformly integrable, then total variation convergence implies convergence in expectation so that,

$$\rho = \lim_{n\to\infty} \frac{m_n}{n} = \lim_{n\to\infty}\mathbb{E}[Y_n] = \mathbb{E}[Y] = \lambda.$$

The precise definition of this uniform integrability is,

$$\lim_{K\to\infty} \sup_{n \in \mathbb{N}} \mathbb{E}[Y_n\mathbb{I}_{Y_n > K}] = \lim_{K\to\infty} \sup_n \frac{1}{n}\sum_{i=1}^n \eta_i\mathbb{I}_{\eta_i > K} = 0.$$

I tried to come up with a nice counterexample, but I can't think of one where $$Q_n$$ are not uniformly integrable and it converges to a measure $$q$$ in total variation. I'd be interested to see if anyone has such a counter example. I suspect that for most if not all counterexamples, $$\rho = \infty$$ and $$\lambda < \infty$$.

• Thank you very much for your elaboration! Actually I am wondering if you are interested in another question of mine (I attached at least twice the bounty and yet until now there is still no good answer to it) math.stackexchange.com/questions/3782421/… Dec 21 '20 at 1:14
• Interesting question. Honestly my pde's need some work. I'll probably attempt it, but no promises. Dec 21 '20 at 4:43