Which theorem did Turing prove in his 1935 King's College dissertation?

In his 1935 King’s College fellowship dissertation, Alan Turing proved a version of the central limit theorem. His dissertation was titled “On the Gaussian Error Function.”

The central limit theorem explains why sums or averages of many independent random quantities often approach a normal, bell-shaped distribution, even when the individual quantities are not normally distributed. Turing’s work investigated this probabilistic behaviour through the mathematics of the Gaussian error function.

There is an important historical qualification. Turing developed the result independently and did not initially know that Jarl Waldemar Lindeberg had proved a closely related theorem in 1922. Turing’s dissertation was nevertheless valued for its originality and technical strength, and it helped him win election to a fellowship at King’s College. It was not his later Princeton doctoral thesis, “Systems of Logic Based on Ordinals,” which concerned ordinal logic and computation.