Question

Breaking stuff let {xi} n i=1 be a sample of random variables of size n with the property that xi ∼ n (µ, σ2 ) and cov(xi , xj ) = ρσ for all pairs i, j.
a.solve for the mean and variance of x¯.
b.what happens to the variance of x¯ as n → ∞?
c.why do the clt and lln fail in this situation?
d.using the provided code, simulate s = 1000 values of x¯ for n = 50, 100, 500. in all three cases calculate the mean and variance of the resulting distribution. create histograms for each. in this code, σ = 1 and ρ = .4. how does the variance for each n compare to the formula from part (a)?

Answer 1

With

$\bar X=\displaystyle\sum_{i=1}^n\frac{X_i}n$

we find the mean is

$\mathrm E[\bar X]=\displaystyle\frac1n\sum_{i=1}^n\mathrm E[X_i]=\frac{n\mu}n=\mu$

and the variance is

$\mathrm V[\bar X]=\displaystyle\frac1{n^2}\mathrm V\left[\sum_{i=1}^nX_i\right]$

$\mathrm V[\bar X]=\displaystyle\frac1{n^2}\left(\sum_{i=1}^n\mathrm V[X_i]+2\sum_{1\le i$

$\mathrm V[\bar X]=\displaystyle\frac1{n^2}\left(n\sigma^2+2\frac{n(n-1)}2\rho\sigma\right)$

$\mathrm V[\bar X]=\dfrac{\sigma^2}n+\dfrac{n-1}n\rho\sigma$

Then as $n\to\infty$, we find that $\mathrm V[\bar X]\to\rho\sigma$. Simply put, the failure of the CLT/LLN has to do with the fact that the $X_i$ are not independent.