Stat 5101 Lecture Notes - School of Statistics

174 Stat 5101 (Geyer) Course Notes6.1.9 The Cauchy DistributionThe Cauchy location-scale family, abbreviated Cauchy(µ, σ) is described inSection B.2.7 of Appendix B an addition rule given by (C.11) in Appendix C,which we repeat hereX 1 + ···+X n ∼Cauchy(nµ, nσ) (6.8)from which we can derive the distribution of the sample meanX n ∼ Cauchy(µ, σ) (6.9)(Problem 6-1).The Cauchy family is not a useful model for real data, but it is theoreticallyimportant as a source of counterexamples. A Cauchy(µ, σ) distribution hascenter of symmetry µ. Hence µ is the median, but µ is not the mean becausethe mean does not exist.The rule for the mean (6.9) can be trivially restated as a convergence indistribution resultX nD−→ Cauchy(µ, σ), as n →∞ (6.10)a “trivial” result because X n actually has exactly the Cauchy(µ, σ) distributionfor all n, so the assertion that is gets close to that distribution for large n istrivial (exactly correct is indeed a special case of “close”).The reason for stating (6.10) is for contrast with the law of large numbers(LLN), which can be stated as follows: if X 1 , X 2 , ... are i. i. d. from a distributionwith mean µ, thenX nP−→ µ as n →∞ (6.11)The condition for the LLN, that the mean exist, does not hold for the Cauchy.Furthermore, since µ does not exist, X n cannot converge to it. But it is conceivablethatP−→ c as n →∞ (6.12)X nfor some constant c, even though this does not follow from the LLN. The result(6.10) for the Cauchy rules this out. Convergence in probability to a constant isthe same as convergence in distribution to a constant (Theorem 2 of Chapter 5in Lindgren). Thus (6.12) and (6.10) are contradictory. Since (6.10) is correct,(6.12) must be wrong. For the Cauchy distribution X n does not converge inprobability to anything.Of course, the CLT also fails for the Cauchy distribution. The CLT impliesthe LLN. Hence if the CLT held, the LLN would also hold. Since the LLNdoesn’t hold for the Cauchy, the CLT can’t hold either.