MHB Converging almost surely proof

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To prove that if Xn converges almost surely to X, then f(Xn) converges almost surely to f(X), it is essential to clarify the definitions of Xn, X, and the function f, including its domain and codomain. The discussion suggests that continuity of the function f may be a sufficient condition for the desired convergence. However, there is an interest in exploring whether this condition can be relaxed. The conversation emphasizes the importance of understanding the properties of the function and the nature of convergence in this context. Overall, the proof hinges on the relationship between the convergence of random variables and the behavior of functions applied to them.
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How do I prove that given Xn converges almost surely to X, that f(Xn) will converge almost surely to f(X)?
 
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Re: converging almost surely proof

Hello oyth! Can you please clarify what is $X_n$, $X$ and what are the domain and codomain of the map $f$? :) Although intuition tells me that probably continuity guarantees what you want, perhaps we can weaken that condition.

Cheers! :D
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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