WIAS Preprint No. 324, (1997)

A nonstandard Hungarian construction for partial sums


  • Grama, Ion G.
  • Nussbaum, Michael

2010 Mathematics Subject Classification

  • 60F17 62G07 60F99


  • Komlós-Major-Tusnády inequality, partial sum process, non-identically distributed variables, function classes, asymptotic equivalence of statistical experiments




We develop a Hungarian construction for the partial sum process of independent, non-identically distributed random variables. The process is indexed by functions ƒ from a functional class 𝓗, but the supremum over ƒ ∈ 𝓗 is taken outside the probability. This nonstandard form is a prerequisite for the functional Komlós-Major-Tusnády inequality in the space of bounded functionals 𝑙 (𝓗), but contrary to the latter it essentially preserves the classical 𝑛-1/2log 𝑛 approximation rate over large functional classes 𝓗 such as the Hölder ball of smoothness 1/2. The nonstandard form has a specific statistical application in the asymptotic equivalence theory for experiments.

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