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Self-normalized central limit theorem with a second-moment denominator

For independent identically distributed

Z_{i}

with mean zero and variance one,

\frac{n ^{- 1/2} \sum _{i = 1}^{n} Z _{i}}{n ^{- 1} \sum _{i = 1}^{n} Z _{i}^{2}} d N (0, 1) .

The numerator obeys the central limit theorem, the denominator converges almost surely to one by the strong law, and Slutsky's theorem combines them.

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