part-ii/2020-123/

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Part II 2020 Probability and Measure: Paper 4, Section II, K

https://questions.tripos.org/part-ii/2020-123/

[cosmic-tarantula]

Spoiler warning

(a) $\mathbb{E}\left[S_n^4\right]$
(b) (i) completeness of $L^2$. $L^2$ convergence implies $L^1$ convergence implies convergence in measure implies weak convergence = convergence in law
(ii)
(iii) characteristic function and Levy's criterion.