Convergence of non increasing sequence of random number

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Discussion Overview

The discussion centers on the convergence properties of a non-increasing sequence of random variables, specifically whether such a sequence, bounded below by a constant \( c \), will converge to \( c \) almost surely. The scope includes theoretical considerations in probability and analysis of sequences.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether a non-increasing sequence of random variables \( \{Y_n\} \), bounded below by a constant \( c \), will converge to \( c \) almost surely.
  • Another participant suggests considering the implications of the sequence being bounded below by \( c-1 \), implying that the choice of lower bound affects the convergence behavior.
  • A further reply clarifies that being bounded below by \( c \) does not provide complete information, as it could also be bounded below by any number less than \( c \), thus raising questions about the distinction between "lower bound" and "greatest lower bound."

Areas of Agreement / Disagreement

Participants express differing views on the implications of the sequence being bounded below by \( c \) and the relevance of other bounds like \( c-1 \). The discussion remains unresolved regarding the convergence of the sequence.

Contextual Notes

There is a lack of clarity regarding the definitions of lower bounds and greatest lower bounds, which may affect the interpretation of convergence in this context.

ensei
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I have a non-increasing sequence of random variables \{Y_n\} which is bounded below by a constant c, \forall \omega \in \Omega. i.e \forall \omega \in \Omega, Y_n \geq c, \forall n. Is it true that the sequence will converge to c almost surely?

Thanks
 
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Hint: If c is such a constant, what about c-1?
 
mfb said:
Hint: If c is such a constant, what about c-1?

All the elements of the sequence are bounded below by c. So, I am not sure what are you trying to say. can you please elaborate?
 
His point is that if the set is bounded below by c, it is also bounded below by c-1 or, for that matter any number less than c. Just saying "bounded below by c" does NOT tell you very much. You seem to be confusing "lower bound" with "greatest lower bound".
 

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