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Sub-exponential Random Variables

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  • Junwei Lu

    (Harvard University)

Abstract

In the previous chapter, we showed the concentration of sample average in the Hoeffding inequality. The asymptotic results like law of large numbers and central limit theorem are also about the sample average. If we look into the proof of these results, we can find that these results rely on the additive formality of the sample mean. So we have the impression that the concentration principle works for the average, but does it cover other statistics? In fact, we have many nonlinear estimators in statistics and machine learning. Can we expect that a general statistic f ( X 1 , … , X n ) $$f(X_1, \ldots, X_n)$$ concentrates to its expectation? The answer is positive. Sample mean is not special.

Suggested Citation

  • Junwei Lu, 2025. "Sub-exponential Random Variables," Springer Books,, Springer.
    • Handle: RePEc:spr:sprchp:978-3-032-03161-7_5
    DOI: 10.1007/978-3-032-03161-7_5
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