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I would be much grateful if you could help me with this.

  1. Is the nonexistence of the expected values of Cauchy distribution because of the heavy tails? How can we prove this using calculus or even better using measure theory?
  2. Can we conclude the nonexistence of the expectation of distributions whose tails are heavier than Cauchy?
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The tail of a Cauchy is $x^{−2}$, with the integral of $|x|×x^{−2}$ at infinity being infinite. The same applies to tails of higher power, like $x^{-α}$ with $1<α≤2$.

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