Convergence of non increasing sequence of random number

[ensei]

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

 

[mfb]

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

 

[ensei]

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?

 

[HallsofIvy]

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".

 

[blue_raver22]

for your question. In order to determine if the sequence \{Y_n\} will converge to c almost surely, we need to consider the definition of almost sure convergence. A sequence of random variables \{X_n\} converges to a random variable X almost surely if the probability that X_n does not converge to X is 0, i.e. P(\lim_{n\rightarrow \infty} X_n = X) = 1.

In this case, we have a non-increasing sequence which is bounded below by a constant c. This means that for any \omega \in \Omega, the values of Y_n will always be greater than or equal to c. This also means that the sequence will not decrease to a value below c, as it is bounded below by c.

Since the sequence is non-increasing and bounded below, we can say that it is monotone and therefore, by the Monotone Convergence Theorem, the sequence will converge to a limit, denoted by L. However, this limit may not necessarily be equal to c. It could be any value between c and the infimum of the sequence, depending on the specific values of the random variables in the sequence.

Therefore, we cannot say for certain that the sequence will converge to c almost surely. It may converge to a value slightly above c, but it could also converge to a value below c. The only thing we can say for sure is that the sequence will converge to a limit L which is bounded below by c.

In order to determine if the sequence will converge to c almost surely, we would need to know more information about the specific values of the random variables in the sequence. We would need to analyze the probabilities of the sequence converging to values above or below c in order to make a conclusion about almost sure convergence. Without this information, we cannot definitively say that the sequence will converge to c almost surely.