Consider a sequence of i.i.d. subgaussian RVs $\{a_{j}\}^{n-1}_{j = 0}$; is $\{a^2_{j}\}^{n-1}_{j = 0}$ uniformly integrable (UI)?
Intuitively it appears to be so; if we take for example $a_j$ i.i.d. symmetric Bernoulli (which is subgaussian) then UI of $a^2_j$ is easy to show since the expectation $\mathbf{E}[|a^2_j|] < +\infty$ and by applying directly the definition of UI in Billingsley, "Probability and Measure", (16.21), p. 220 the result is immediate.
Actually, in these notes example 6 claims that all zero-mean, unit-variance RVs are UI. However, the generalization to the initial statement of this question does not seem immediate, i.e., is it true that the square of a subgaussian RV is uniformly integrable?
Since this question arises in an engineering application of a particular CLT and I must keep things as simple as possible, I would like to know if anyone is aware of a quotable result in this direction, or if this statement is trivial enough that a reference to the definition of uniform integrability suffices.
Uniform integrability is concerned only with the distributions of the random variables hence every collection of i.i.d. random variables is equivalent to any single random variable in the collection, as far as uniform integrability is concerned.
In particular every collection of i.i.d. integrable random variables is uniformly integrable because a single integrable random variable is uniformly integrable.