The sample mean of a Normal RV does tend towards the population expected value, sure.
The GP comment is talking about a Cauchy RV, which has heavy tails. So it has enough probability mass at large values that the expected value is infinite. Discarding constant scale factors, in this case:
E[X] = Int{0..infty} x * p(x) dx
= Int{0..infty} x * (1/x^2) dx
= Int{0..infty} (1/x) dx
= +infty
So, the sample mean of Cauchy random variates will not converge to any real number.
The GP comment is talking about a Cauchy RV, which has heavy tails. So it has enough probability mass at large values that the expected value is infinite. Discarding constant scale factors, in this case:
So, the sample mean of Cauchy random variates will not converge to any real number.