Question

Problem 12.6 (Optional: Strong Law of Large Number). Let $X_1, X_2, \dots$ be a iid random sequence with $E[X_i] = 0$, $E[X_i^2] = \sigma^2$, and $E[|X_i|^4] < \infty$. Consider the sample mean \begin{equation} \overline{X}_n := \frac{1}{n} \sum_{i=1}^n X_i. \end{equation} (i) Show that $\overline{X}_n \stackrel{a.s.}{\rightarrow} 0$. Hint: You may use the fact that $E[\overline{X}_n^4] = \frac{n\gamma + 3n(n-1)\sigma^4}{n^4}$.

          Problem 12.6 (Optional: Strong Law of Large Number). Let $X_1, X_2, \dots$ be a iid random sequence with $E[X_i] = 0$, $E[X_i^2] = \sigma^2$, and $E[|X_i|^4] < \infty$. Consider the sample mean \begin{equation*} \overline{X}_n := \frac{1}{n} \sum_{i=1}^n X_i. \end{equation*} (i) Show that $\overline{X}_n \stackrel{a.s.}{\rightarrow} 0$. Hint: You may use the fact that $E[\overline{X}_n^4] = \frac{n\gamma + 3n(n-1)\sigma^4}{n^4}$.

Problem 12.6 (Optional: Strong Law of Large Number). Let X1, X2, … be a iid random sequence with E[Xi] = 0, E[Xi^2] = σ^2, and E[|Xi|^4] < ∞. Consider the sample mean
Xn := (1)/(n)∑i=1^n Xi.
(i) Show that Xn a.s.→ 0. Hint: You may use the fact that E[Xn^4] = (nγ + 3n(n-1)σ^4)/(n^4).

Added by Randy H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Jacob Fry

11/30/2022

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.. with finite mean μ, the sample mean Xn converges almost surely to μ as n approaches infinity. Show more…

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Problem 12.6 Optional: Strong Law of Large Numbers. Let X1, X2, ... be an iid random sequence with E[X]=0, E[X]= and E[X]=<. Consider the sample mean Xn: Xn = (1/n) * Σ(i=1 to n) Xi Show that Xn → 0. Hint