Thread: Sum of Cauchy Distributions

Hi - Looking to see if there is a simple way to prove that if X1 is Cauchy(mu1, sigma1), and X2 is Cauchy(mu2,sigma2), X1 and X2 are indep, then X1+X2 is Cauchy with mu1+mu2 and sigmal1+sigma2.

Seems like a bear to go through the partial fraction decomp of the convolution density

Thanks much

Ravi

Originally Posted by raramasw

Hi - Looking to see if there is a simple way to prove that if X1 is Cauchy(mu1, sigma1), and X2 is Cauchy(mu2,sigma2), X1 and X2 are indep, then X1+X2 is Cauchy with mu1+mu2 and sigmal1+sigma2.

Seems like a bear to go through the partial fraction decomp of the convolution density

Thanks much

Ravi

Try using a moment-generating function approach.

MGF for Cauchy does not exist. Characteristic function does, and Feller V2 mentions something about using Fourier Analysis to prove this. No details

Originally Posted by raramasw

Hi - Looking to see if there is a simple way to prove that if X1 is Cauchy(mu1, sigma1), and X2 is Cauchy(mu2,sigma2), X1 and X2 are indep, then X1+X2 is Cauchy with mu1+mu2 and sigmal1+sigma2.

Seems like a bear to go through the partial fraction decomp of the convolution density

Thanks much

Ravi

The characteristic function of the distribution of the sum of independent RVs is the product of their characteristic functions. Look up the CF of the Cauchy distributions multiply them and you will find that it is of the same form as a Cauchy distribution with the required properties.

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Originally Posted by raramasw

MGF for Cauchy does not exist. Characteristic function does, and Feller V2 mentions something about using Fourier Analysis to prove this. No details

A slip of the keyboard .... The fingers don't always type what the brain is thinking (nevertheless it looks like you got put on the right track). Thanks for the catch.