Cramér's large deviation theorem
Imagine you have a bunch of marbles in a box. You know that most of the time when you reach in the box to grab a marble, you will get a certain color, like blue. But sometimes, every once in a while, you might get a different color, like red or green. And when you do get a different color, it's not all that surprising, because it doesn't happen very often.
Now imagine that instead of marbles in a box, you have a bunch of numbers, and you want to know how often certain numbers show up. Cramér's large deviation theorem helps you understand how likely it is to get a number that falls outside of the usual range.
So basically, if you're looking at a bunch of numbers and you want to know how often you might see a number that's really different from the others, this theorem can help you figure it out. It can tell you how likely it is, and how far away from the usual range that number might be. This can be really helpful if you're trying to do things like predict stock prices or understand the behavior of complex systems.
Now imagine that instead of marbles in a box, you have a bunch of numbers, and you want to know how often certain numbers show up. Cramér's large deviation theorem helps you understand how likely it is to get a number that falls outside of the usual range.
So basically, if you're looking at a bunch of numbers and you want to know how often you might see a number that's really different from the others, this theorem can help you figure it out. It can tell you how likely it is, and how far away from the usual range that number might be. This can be really helpful if you're trying to do things like predict stock prices or understand the behavior of complex systems.