Central Limit Theorem

The central limit theorem is a result in probability theory which states that when many independent random values drawn from the same distribution are added or averaged, the result is approximately normally distributed, provided the underlying distribution has a finite variance. The original values need not be normal. They may be skewed, flat or discrete; the bell shape emerges in the sum.

The theorem explains why the normal distribution appears so often in measurement and statistics, and it underpins most methods for judging how reliable an estimate is. In finance it justifies the standard tests applied to average returns, the square-root rule for scaling risk over time, and the way simulation results are assessed. Its conditions are not always met by market data, and knowing where it fails is as important as knowing what it says.

Let $X_1, X_2, \dots, X_n$ be independent values from a distribution with mean $\mu$ and standard deviation $\sigma$, and let $\bar{X}$ be their average. For large $n$, the average is approximately normal with mean $\mu$ and a standard deviation, called the standard error, of:

Equivalently, the standardized average approaches the standard normal distribution:

• $\bar{X}$ is the sample mean of the $n$ values.
• $\mu$ is the true mean of the distribution, the population mean.
• $\sigma$ is the true standard deviation.
• $n$ is the number of values averaged.

The theorem is different from the law of large numbers. The law says the sample mean settles on the true mean as $n$ grows. The central limit theorem adds the shape and size of the remaining error.

A trading strategy has been run for 250 days. Its daily returns average 0.05 percent with a standard deviation of 1 percent. The question is whether the positive average is real or could be luck.

The standard error of the mean is 1 ÷ √250 = 1 ÷ 15.81 = 0.063 percent. The observed mean is 0.05 ÷ 0.063 = 0.79 standard errors above zero. Under a normal distribution, a value that far from zero or further arises by chance more than 40 percent of the time. The evidence for a true positive return is weak, even though 0.05 percent a day would compound to about 13 percent a year. With the same figures over 2,500 days, the standard error falls to 0.02 percent and the mean is 2.5 standard errors from zero, which is much stronger evidence.

The first version was found by Abraham de Moivre in 1733, who showed that the binomial distribution for a large number of coin tosses approaches the bell curve. Pierre-Simon Laplace generalized the result in the early nineteenth century. Rigorous proofs under broad conditions came from Russian mathematicians, including Aleksandr Lyapunov, around 1900. The name "central limit theorem" was introduced by George Pólya in 1920, the word central referring to its importance in probability theory.

• Testing performance. A t-statistic for a strategy's mean return, and the confidence interval around it, both rely on the sample mean being approximately normal. The whole apparatus of hypothesis testing on returns rests on the theorem.
• Scaling over time. If daily returns are independent, the return over many days is a sum, with a standard deviation that grows with the square root of the number of days. Long-horizon returns are closer to normal than daily ones.
• Simulation. The error in a Monte Carlo simulation shrinks with the square root of the number of trials, so quadrupling the trials halves the error.
• Diversification. The risk of an average of many independent bets falls as their number rises, which is the statistical basis for spreading capital across many small positions.
• Continuous-time models. Brownian motion arises as the limit of a random walk with many small steps, an application of the same theorem.

The theorem has conditions, and financial data strains each of them. First, the variance must be finite. Returns have fat tails, and for very heavy-tailed distributions convergence to the normal is slow or does not occur. Second, the values must be independent, or nearly so. Returns show little correlation from day to day, but their volatility clusters, so the size of one day's move helps predict the size of the next. Dependence of this kind makes the effective sample smaller than it appears.

Third, the approximation is best near the center of the distribution and worst in the tails. An average of 250 daily returns may look normal around its mean while extreme outcomes remain far more likely than the normal curve suggests. Risk measures that depend on the tails, such as value at risk at high confidence levels, are therefore understated by a normal assumption. Finally, positions in a portfolio are rarely independent. In a crisis, correlations rise and the diversification the theorem appears to promise is reduced when it is most needed.