Convergence of non increasing sequence of random number

In summary, the conversation discusses the convergence of a non-increasing sequence of random variables that is bounded below by a constant c for all possible outcomes. The question raised is whether this sequence will converge to c almost surely. There is a discussion about the meaning of being bounded below by c and how it relates to the greatest lower bound.
  • #1
ensei
2
0
I have a non-increasing sequence of random variables [itex] \{Y_n\}[/itex] which is bounded below by a constant [itex]c[/itex], [itex]\forall \omega \in \Omega[/itex]. i.e [itex]\forall \omega \in \Omega[/itex], [itex]Y_n \geq c[/itex], [itex]\forall n[/itex]. Is it true that the sequence will converge to [itex]c[/itex] almost surely?

Thanks
 
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  • #2
Hint: If c is such a constant, what about c-1?
 
  • #3
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?
 
  • #4
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".
 
  • #5
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.
 

Related to Convergence of non increasing sequence of random number

1. What is a non-increasing sequence of random numbers?

A non-increasing sequence of random numbers is a sequence of numbers where each number is less than or equal to the previous number. This means that the sequence is decreasing or staying the same as it progresses.

2. How is the convergence of a non-increasing sequence of random numbers determined?

The convergence of a non-increasing sequence of random numbers is determined by analyzing the behavior of the sequence as the number of terms increases. If the sequence approaches a specific number or value as the number of terms increases, it is said to converge. If the sequence does not approach a specific number, it is said to diverge.

3. What are the implications of a non-converging non-increasing sequence of random numbers?

If a non-increasing sequence of random numbers does not converge, it means that the sequence is unpredictable and does not approach a specific number or value. This can make it difficult to make predictions or draw conclusions based on the sequence.

4. How is the convergence rate of a non-increasing sequence of random numbers determined?

The convergence rate of a non-increasing sequence of random numbers is determined by how quickly the sequence approaches a specific number or value. A faster convergence rate means that the sequence approaches the number or value at a faster rate, while a slower convergence rate means that the sequence approaches the number or value at a slower rate.

5. What are some real-world applications of studying the convergence of non-increasing sequences of random numbers?

The study of the convergence of non-increasing sequences of random numbers has various applications in fields such as finance, economics, and statistics. It can help in predicting stock market trends, analyzing economic data, and understanding the behavior of random processes.

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