Almost Sure Convergence
| Example 1 | |
|---|---|
| Consider an animal of some short-lived species. We record the amount of food that this animal consumes per day. This sequence of numbers will be unpredictable, but we may be quite certain that one day the number will become zero, and will stay zero forever after. | |
| Example 2 | |
| Consider a man who tosses seven coins every morning. Each afternoon, he donates one pound to a charity for each head that appeared. The first time the result is all tails, however, he will stop permanently. Let X1, X2, … be the daily amounts the charity receives from him. We may be almost sure that one day this amount will be zero, and stay zero forever after that. However, when we consider any finite number of days, there is a nonzero probability the terminating condition will not occur. |
This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis.
Read more about this topic: Convergence Of Random Variables