Continuous Mapping Theorem

Definition

The Continuous Mapping Theorem states that if a sequence of random variables converges in distribution to a random variable and a function is continuous, then the transformed sequence of random variables will also converge in distribution to the transformed random variable. This theorem is crucial in understanding how functions of random variables behave under convergence, connecting the properties of convergence in probability and the continuity of functions.

5 Must Know Facts For Your Next Test

1. The Continuous Mapping Theorem is particularly useful when dealing with transformations of random variables, allowing statisticians to understand how these transformations behave under convergence.
2. This theorem applies specifically to convergence in distribution, meaning that it doesn't necessarily hold for other types of convergence, such as convergence in probability.
3. If a sequence of random variables converges to a constant, any continuous function applied to that sequence will converge to the function's value at that constant.
4. The theorem highlights the importance of continuity; if the function applied is not continuous at the limit point, then convergence may not hold.
5. In applications, this theorem is often used to derive the limiting distributions of statistics derived from estimators or test statistics.

Review Questions

• How does the Continuous Mapping Theorem connect convergence in distribution with continuous functions?
  • The Continuous Mapping Theorem establishes a clear link between convergence in distribution and continuous functions by stating that if a sequence of random variables converges in distribution to some limit and a function is continuous, then applying that function to the sequence will result in a new sequence that also converges in distribution. This means that when you take a limit in distribution, you can apply continuous transformations without losing the property of convergence.
• What implications does the Continuous Mapping Theorem have when dealing with functions that are not continuous at certain points?
  • When dealing with functions that are not continuous at certain points, the Continuous Mapping Theorem does not guarantee that convergence will hold. If the sequence of random variables approaches a limit where the function is discontinuous, then the transformed sequence may fail to converge to the expected limit. This shows that continuity at the limit point is essential for applying the theorem correctly and ensuring that we retain convergence properties after transformation.
• Evaluate how the Continuous Mapping Theorem can be utilized in statistical inference and provide an example.
  • The Continuous Mapping Theorem is utilized in statistical inference when determining the limiting behavior of estimators or test statistics under transformations. For example, if we have an estimator for a parameter that converges in distribution to a normal distribution as sample size increases, we can use the theorem to show that any continuous transformation of this estimator, such as taking its square or applying another function, will also converge to a normal distribution. This is fundamental when constructing confidence intervals or hypothesis tests involving transformations.

Related terms

Convergence in Distribution: A type of convergence where the cumulative distribution functions of a sequence of random variables converge to the cumulative distribution function of a limiting random variable at all points where the limiting function is continuous.

Continuous Function: A function is continuous if small changes in the input result in small changes in the output, meaning there are no abrupt jumps or breaks in its graph.

Random Variable: A variable whose possible values are outcomes of a random phenomenon, typically characterized by a probability distribution.