Discussion
Let \(\{X_{n}\}\) be a sequence of random variables and X a random variable such that :
\[ \sum_{n=1}^{\infty} P[|X_{n} - X| \gt \epsilon] \lt \infty \]
for every \(\epsilon \gt 0\).
Consider the statements :
(I) \(P\left[\lim_{n \to \infty} X_{n} = X\right] = 1\)
(II) \(P\left[\lim_{n \to \infty} X_{n}^{2} = X^{2}\right] = 1\)
(III) \(P\left[\lim_{n \to \infty} X_{n} = X\right]\) only if \(\{X_{n}\}\) is a sequence of independent random variables.
Which of the above are always correct ?
Answer: B
The statements always correct are (I) and (II) only.
By the Borel-Cantelli lemma, \(\sum P[|X_{n} - X| \gt \epsilon] \lt \infty\) for every \(\epsilon\) means that, almost surely, \(|X_{n} - X| \gt \epsilon\) happens only finitely often. So \(X_{n} \to X\) almost surely (I). Since \(x \mapsto x^{2}\) is continuous, \(X_{n}^{2} \to X^{2}\) almost surely too (II).
(III) is false: this half of Borel-Cantelli needs no independence. Independence matters only for the converse, the second Borel-Cantelli lemma. So options including (III), or 'none of the three', are wrong.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development