Continuous Mapping Theorem

Definition

The Continuous Mapping Theorem states that if a sequence of random variables converges in distribution to a limit, and a continuous function is applied to these variables, then the transformed variables will also converge in distribution to the function applied to the limit. This theorem highlights the interplay between convergence types and the effect of continuous functions on these convergences.

5 Must Know Facts For Your Next Test

1. The Continuous Mapping Theorem applies specifically to convergence in distribution, which is essential for understanding how functions affect random variables.
2. If a sequence of random variables {X_n} converges in distribution to X, and g is continuous, then g(X_n) converges in distribution to g(X).
3. This theorem is particularly useful for transforming limits of random variables when dealing with statistical inference and asymptotic analysis.
4. The Continuous Mapping Theorem can also be applied to multi-dimensional cases where functions are continuous in multiple variables.
5. It establishes a crucial link between pointwise convergence of sequences and their behavior under continuous transformations.

Review Questions

• How does the Continuous Mapping Theorem relate convergence in distribution to continuous functions?
  • The Continuous Mapping Theorem establishes that if a sequence of random variables converges in distribution to a limit, applying a continuous function to those random variables will also yield convergence in distribution of the transformed variables. This relationship shows how continuity preserves convergence properties and helps in analyzing the behavior of random variables when subjected to transformations.
• Discuss an example where the Continuous Mapping Theorem is applied to demonstrate convergence in distribution using a specific continuous function.
  • Consider the sequence of random variables defined as X_n = n^{-1}Z, where Z follows a standard normal distribution. As n increases, X_n converges in distribution to 0. By applying the continuous function g(x) = x^2, we observe that g(X_n) = (n^{-1}Z)^2 converges in distribution to g(0) = 0. This example illustrates how using a continuous function affects the convergence properties of the original sequence.
• Evaluate the implications of the Continuous Mapping Theorem on statistical methods involving limits and transformations.
  • The Continuous Mapping Theorem has significant implications for statistical methods, especially when dealing with asymptotic distributions and estimators. It allows researchers to confidently apply transformations to estimators or test statistics without losing control over their limiting behavior. Consequently, this theorem enhances the robustness of statistical inference techniques, enabling analysts to derive results from limiting distributions while ensuring that continuous transformations maintain desired convergence properties.

Related terms

Convergence in Distribution: A type of convergence where a sequence of random variables approaches a limiting distribution as the sample size increases.

Continuous Function: A function that does not have any abrupt changes or jumps in its value, meaning small changes in the input result in small changes in the output.

Random Variable: A variable whose possible values are numerical outcomes of a random phenomenon, often described by a probability distribution.