Contraction principle (large deviations theory)
Okay kiddo, have you ever played with two pieces of clay? When you push them together, they become one big piece of clay, right? The contraction principle is kind of like that, but it's not about clay.
It's a math idea that helps us understand how things change over time. Imagine we have a machine that takes in one number and gives us another number as a result. We can think of this machine as a function. Now we start with any number and keep using the machine over and over again. What happens to the numbers we get as we repeat the process?
The contraction principle says that if the function we're using has a special property, the numbers we get will get closer and closer together the more times we use the machine. It's like squishing the clay together more and more until it's all one piece.
This property is really helpful because it helps us understand probabilities. Sometimes we want to know how likely it is that something will happen, but it's hard to calculate exactly. Using the contraction principle, we can get a good estimate of how likely something is to happen even if we can't calculate it exactly.
So, the contraction principle is like a math machine that squishes numbers closer and closer together over time, just like how we can squish two pieces of clay together. It helps us estimate the probability of things happening too.
It's a math idea that helps us understand how things change over time. Imagine we have a machine that takes in one number and gives us another number as a result. We can think of this machine as a function. Now we start with any number and keep using the machine over and over again. What happens to the numbers we get as we repeat the process?
The contraction principle says that if the function we're using has a special property, the numbers we get will get closer and closer together the more times we use the machine. It's like squishing the clay together more and more until it's all one piece.
This property is really helpful because it helps us understand probabilities. Sometimes we want to know how likely it is that something will happen, but it's hard to calculate exactly. Using the contraction principle, we can get a good estimate of how likely something is to happen even if we can't calculate it exactly.
So, the contraction principle is like a math machine that squishes numbers closer and closer together over time, just like how we can squish two pieces of clay together. It helps us estimate the probability of things happening too.