In this master’s thesis, I examine the behavior of the sum of random variables, assuming they are independent and identically distributed. To this end, I make use of fundamental probability theorems and inequalities, such as Markov’s and Chebyshev’s inequalities and the Central Limit Theorem. In addition, I analyze extensions of the classical CLT, including the Berry-Esseen theorem and large deviation theory. To better illustrate the findings, I apply the results to the context of roulette gambling. Using the example of a player who consistently bets using the same strategy across multiple rounds, I investigate how likely he is to achieve a positive return under different strategies.
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