Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does this article make?
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This muddled jumble actually refers to three different phenomena!
If we take the article to mean "beware of extrapolating previous growth into the future," we will get much out of it. Investors who think this company is a good buy because its PE ratio is low (which includes several of the notable money managers quoted in the article) are no better than the "ponderous scientific people" Twain skewered over a century ago.
A better acquaintance with Bernoulli, Hotelling, and Twain would have improved the accuracy and readability of this article, but in the end it seems to have gotten the message right.
Humorously enough, I just wrote a blog post on this very subject: http://confounding.net/2012/03/12/thats-not-how-the-law-of-large-numbers-works/
Essentially, the Law of Large Numbers is that as the number of trials of a random process increases, the mean of those trials will approach the actual mean (or expectation, for more complex distributions). So while if you flip a coin once and get heads your probability of heads = 1.0, as you flip more and more coins, you'll head closer and closer to 0.50.
The author argues that Apple will have trouble in the future due to something that is not actually at all related to the Law of Large Numbers. Namely, that as Apple grows larger, the same % increase in share price, earnings, etc. get harder to reach in absolute dollar terms. Basically, to stay on course, Apple has to get larger and larger hits.
Linking that to the behavior of a random process converging to a mean requires some serious mental gymnastics. As far as I can tell, the assertion is that "The awesomeness of your products" is a random process, and while Apple has had a streak of "Above Average" awesome, they'll eventually have to converge toward a mean of "Middling". But that's being really charitable to the author.
Just because 500 billion is a large number doesn't mean the "Law of Large Numbers" is what's acting on it.
There is no reason to think that stock price draws over time for a particular company represent independent, identically distributed random variables.