Weak convergence of the sum of dependent variables, question

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Discussion Overview

The discussion revolves around the weak convergence of the sum of dependent random variables, specifically examining whether a counterexample can be constructed when independence between the variables does not hold. The scope includes theoretical aspects of probability and convergence in distribution.

Discussion Character

  • Exploratory
  • Debate/contested

Main Points Raised

  • One participant presents a problem involving weak convergence of sums of random variables and asks for a counterexample when independence is not assumed.
  • Another participant suggests using Yn = -Xn as a potential counterexample.
  • A subsequent reply argues that Yn = -Xn does not serve as a counterexample since it leads to a trivial convergence to zero, which does not satisfy the conditions of the original problem.
  • Further clarification is provided that weak convergence could be maintained by setting Y as independent and identically distributed (iid) to -X instead of simply Y = -X, which would allow for non-trivial sums.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the proposed counterexample. There are competing views on what constitutes a valid counterexample in the context of weak convergence without independence.

Contextual Notes

The discussion highlights the nuances of weak convergence and the implications of independence on the behavior of sums of random variables. There are unresolved aspects regarding the conditions under which weak convergence holds and the definitions of independence in this context.

vovchik
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Hi guys,

Problem: Let {Xn},{Yn} - real-valued random variables.
{Xn}-->{X} - weakly; {Yn}-->{Y} weakly.
Assume that Xn and Yn - independent for all n and that X and Y - are independent.
Fact that {Xn+Yn}-->{X+Y} weakly, can be shown using characteristic functions and Levy's theorem.

Question:
If independence does not hold, can you construct a counterexample?

I appreciate any help in advance.
 
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How about simply Yn = -Xn?
 
bpet said:
How about simply Yn = -Xn?

This is not a counterexample Xn -> X, Yn -> Y (=-X) Xn + Yn -> 0 (= X + Y).
 
mathman said:
This is not a counterexample Xn -> X, Yn -> Y (=-X) Xn + Yn -> 0 (= X + Y).

With weak convergence you could set Y iid to -X instead of Y=-X (so that X+Y <> 0 if X is non-trivial).
 

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