The Lindeberg-Feller Central Limit Theorem

A trading desk's daily P&L is the sum of many positions: a large S&P future, a handful of single-name options, a small basis trade. The ordinary Central Limit Theorem promises that a sum of many identical, independent shocks looks normal. But these positions are nothing alike in size — one line can be ten times the risk of another. Does the sum still settle into a bell curve, or can one oversized position wreck the average for everyone else?

The analogy: one loud voice in a choir

Imagine averaging the volumes of a hundred singers to predict the chorus's overall loudness. If everyone sings at roughly the same volume, the average is stable and predictable — that's the ordinary CLT. Now put one singer with a bullhorn in the back row. However quiet the other ninety-nine are, the bullhorn dominates the total, and the sum no longer behaves like an average of "typical" contributors — it behaves like that one voice. Lindeberg-Feller is the precise rule for how loud any single singer is allowed to get, relative to the whole choir, before this happens.

Writing it down

Let $X_1, X_2, \ldots, X_n$ be independent (not necessarily identical) random variables, each with mean zero and variance $\sigma_i^2$. Write $s_n^2 = \sum_{i=1}^n \sigma_i^2$ for the total variance of the sum. The Lindeberg condition asks that for every $\epsilon > 0$,

$$\lim_{n \to \infty} \frac{1}{s_n^2} \sum_{i=1}^n E\left[X_i^2 \, \mathbf{1}_{\{|X_i| > \epsilon s_n\}}\right] = 0.$$

In words: look at the fraction of total variance coming from pieces large enough to be a noticeable slice of the whole sum's spread — and check that fraction shrinks to nothing as you add more pieces. No single term (or small cluster of terms) is allowed to keep contributing a fixed share of the total variance forever. If this holds, then

$$\frac{S_n}{s_n} = \frac{X_1 + \cdots + X_n}{s_n} \xrightarrow{d} N(0,1).$$

In words: the properly rescaled sum converges to a standard normal, even though the pieces have different sizes and different distributions. The ordinary CLT is the special case where all $\sigma_i$ are equal — there, no single term can ever be a large share of the total, so the condition is automatic.

Worked example 1: a diversified book

Ten independent positions, each with variance $\sigma_i^2 = 4$ (so $s_n^2 = 40$ after ten). The largest single variance is 4, only $4/40 = 10\%$ of the total, and that share shrinks toward zero as you add more similar positions. Lindeberg's condition is satisfied in the limit, and the standardized P&L $S_n / \sqrt{40}$ is well approximated by $N(0,1)$. If each position also has bounded, roughly symmetric outcomes (no lottery-ticket payoffs), 40–50 such positions already give a serviceable normal approximation for a 95% VaR estimate.

Worked example 2: one dominant name

Now replace one small position with a concentrated bet: nine positions with $\sigma_i^2 = 1$ each (total 9) and one large position with $\sigma_i^2 = 91$, so $s_n^2 = 100$. That one term supplies $91/100 = 91\%$ of total variance — nowhere near shrinking to zero, no matter how many small positions you bolt on around it. Lindeberg fails. The rescaled sum $S_n / 10$ does not converge to $N(0,1)$; instead it inherits the shape of the dominant position's own distribution. If that position's return is skewed (say, a short-dated option near expiry with a fat left tail), the "portfolio P&L is roughly normal" assumption a risk model quietly relies on is simply false, no matter how many other diversifying trades sit around it.

Run several batches of paths with very different starting notionals. Notice how a single oversized path can visibly drag the average path away from the smooth, symmetric spread the other paths would produce on their own — a live picture of one term dominating the sum.

What this means in practice

• Portfolio risk aggregation. A VaR model that sums many position P&Ls and calls the total normal is implicitly assuming Lindeberg holds. Concentrated books — one massive futures line plus small satellite trades — are exactly where that assumption breaks.
• Position limits. Capping any single position's variance at, say, 10–15% of book variance is a practical way to keep Lindeberg's condition roughly true, and keep the normal approximation trustworthy.
• Stress testing. When Lindeberg fails, don't approximate the tail with a normal quantile — simulate, or model the dominant term's own distribution directly (see The Pareto Distribution and Power Laws for what a dominant fat-tailed term can do to the sum).