Elementary Mathematical Inequalities to Prove the Laws of Large Numbers
DOI:
https://doi.org/10.54097/kgk41x22Keywords:
Statistics, Applied mathematics, Chebyshev’s inequality, Markov’s inequality, Borel–Cantelli lemma, Kolmogorov’s maximal inequality, WLLN, SLLN, LLN.Abstract
This paper gives a simple and clear proof of the Weak Law of Large Numbers (WLLN) and the Strong Law of Large Numbers (SLLN). We only use basic tools from real analysis and elementary probability. First, we prove Markov’s inequality, Cheby- shev’s inequality, and the Borel–Cantelli lemma in a direct way. We also give a proof of Kolmogorov’s maximal inequality. After that, we use these inequalities to prove the WLLN with Chebyshev’s inequality and the SLLN with Kolmogorov’s inequality and the Borel–Cantelli lemma. The main goal of this paper is to show that the laws of large num- bers can be proved with simple ideas, without advanced probability theory. Our work helps beginners understand these important results in a more easy and friendly way.
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