The law of large WTF?

Andrew Gelman recently pointed out that some financial folks have some very strange ideas about the law of large numbers. Apparently, those strange ideas are deeply ingrained. Here are some excerpts from Investopedia's entry on the law of large numbers:

Definition of 'Law of Large Numbers.' In statistical terms, a rule that assumes that as the number of samples increases, the average of these samples is likely to reach the mean of the whole population.

So far so good; it's an informal definition of the same concept that statisticians call the law of large numbers. But then when you read,

When relating this concept to finance, it suggests that as a company grows, its chances of sustaining a large percentage in growth diminish. This is because as a company continues to expand, it must grow more and more just to maintain a constant percentage of growth.

...you start scratching your head. The above statement, while definitely true, has precisely fuck-all to do with the law of large numbers. As does the next one:

As an example, assume that company X has a market capitalization of $400 billion and company Y has a market capitalization of $5 billion. In order for company X to grow by 50%, it must increase its market capitalization by $200 billion, while company Y would only have to increase its market capitalization by $2.5 billion. The law of large numbers suggests that it is much more likely that company Y will be able to expand by 50% than company X.

No matter how you look at it, this is not an example of the law of large numbers. Whoever wrote this entry does not understand the definition they provided.

Added: I missed the mistake in Investopedia's definition of the law: It's not an assumption, it's a theorem.

More Added: For the sake of comparison, here is Felix Salmon's example of a correct use of the law of large numbers in a business setting:

If you run a bunch of casinos with hundreds of thousands of punters coming and and betting hundreds of millions of dollars, then you can predict with high accuracy the amount of money you're going to make at the end of the quarter.