Fall 2011 Strategic Practice 11: Section 1 (Law of Large Numbers, Central Limit Theorem) - Question 3
Let $T_{1},T_{2}, . . .$ be i.i.d. Student-t r.v.s with $m\geq 3$ degrees of freedom. Find constants $a_{n}$ and $b_{n}$ (in terms of m and n) such that $a_{n}(T_{1}+T_{2}+. . . +T_{n}-b_{n})$ converges to N(0, 1) in distribution as $n\rightarrow \infty$.
Solution: Consult iTunes course for full detailed solutions. Find the mean and variance of each Tj using LOTUS. By the Central Limit Theorem CLT (and linearity of E, and the fact that the variance of the sum of uncorrelated r.v.s is the sum of the variances), we find our desired solution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."