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Deriving the Central Limit Theorem from the De Moivre-Laplace Theorem
Calvin Wooyoung Chin
The de Moivre-Laplace theorem is a special case of the central limit theorem for Bernoulli random variables, and can be proved by direct computation. We deduce the central limit th…
A Short and Elementary Proof of the Central Limit Theorem by Individual Swapping
Calvin Wooyoung Chin
We present a short proof of the central limit theorem which is elementary in the sense that no knowledge of characteristic functions, linear operators, or other advanced results ar…
A Gambler that Bets Forever and the Strong Law of Large Numbers
Calvin Wooyoung Chin
In this expository note, we give a simple proof that a gambler repeating a game with positive expected value never goes broke with a positive probability. This does not immediately…
Necessary and Sufficient Conditions for Convergence to the Semicircle Distribution
Calvin Wooyoung Chin
We consider random Hermitian matrices with independent upper triangular entries. Wigner's semicircle law says that under certain additional assumptions, the empirical spectral dist…