Consider the Lindeberg–Feller central limit theorem (CLT), which we state as follows. Let {x_n} be a sequence of independent random variables with means {μ_n} and nonzero variances {σ_n²} (both existing), and c.d.f.s {F_n}. Define λ_n > 0 by

. Then, Lindeberg's condition

is equivalent to

where the latter limit is known as the uniform asymptotic negligibility (u.a.n.) condition. One can usually interpret λ_n² as the variance of the numerator of z_n. We shall see, however, that there are cases where asymptotic normality holds in spite of {x_n} having infinite variances.

Let {x_n} be a random sample from Student's t(ν). For ν < 2, no λ_n exists that can lead to

. This is because the tails of the density of t(ν) decay at a rate of u^−ν−1 and the stable limit theorem tells us that a nonnormal stable law arises if the tails of the p.d.f. of x_i decay at a rate of u^−a where a < 3; e.g., see Loève (1977, §25) or Hoffmann-Jørgensen (1994, §5.25). For example, for ν = 1, the average of standard Cauchy variates is standard Cauchy too, so that there exists no λ_n achieving asymptotic normality of z_n.

For ν > 2, both the mean and the variance exist, and the Lindeberg–Feller CLT applies, with λ_n² = n var(x_i). The interesting part is ν = 2, where we will show that asymptotic normality of z_n holds, in spite of var(x_i) being infinite, and we will derive the appropriate λ_n. We will require the additional assumption that λ_n² → ∞ as n → ∞. In the standard CLT, this assumption was unnecessary, as it followed from λ_n² = n var(x_i). We will see subsequently that λ_n² can be interpreted in terms of truncated variances for ν = 2.

To prove the asymptotic normality of z_n, we need to show that the characteristic function φ(t) := E(exp(itx_i)) satisfies

for some choice of λ_n, with λ_n → ∞ as n → ∞. Because the sequence {x_n} is i.i.d., the uniform asymptotic negligibility condition

is satisfied for all ε > 0, thus implying

This allows us to take the leading term of the logarithmic expansion of the left-hand side of (1) as

where the linear term in t drops out because the sequence {x_n} is centered around zero. Asymptotic standard-normality obtains if we can find the appropriate λ_n² that makes the latter limit equal to 1 for all ε > 0. Notice that this limit is the complement of Lindeberg's condition, where

is replaced by n because {x_n} is an i.i.d. sequence.

From Student's t(2) density,

tends to infinity as c → ∞. We need to solve

where we have dropped

that is dominated by sinh⁻¹(c) → ∞. By using the logarithmic representation of the latter and simplifying,

is solved by

or any other function that is asymptotically equivalent to it (such as

). Therefore,