The Lyapunov central limit theorem is a statistical principle that extends the classical central limit theorem. It states that if you have a sequence of independent random variables with finite variances, the sum of these variables will approximate a normal distribution as the number of variables increases, provided certain conditions are met. This theorem is particularly useful when dealing with non-identically distributed variables. To apply the Lyapunov condition, the random variables must satisfy a specific criterion regarding their moments. This ensures that the influence of any single variable diminishes as the number of variables grows, allowing the overall sum to converge to a normal distribution, which is fundamental in probability theory and statistics.