Limit theorems for Bajraktarevic and Cauchy quotient means of independent identically distributed random variables

Barczy, Mátyás and Burai, Pál József (2022) Limit theorems for Bajraktarevic and Cauchy quotient means of independent identically distributed random variables. AEQUATIONES MATHEMATICAE, 96 (2). pp. 279-305. ISSN 0001-9054

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Abstract

We derive strong laws of large numbers and central limit theorems for Bajraktarevic, Gini and exponential- (also called Beta-type) and logarithmic Cauchy quotient means of independent identically distributed (i.i.d.) random variables. The exponential- and logarithmic Cauchy quotient means of a sequence of i.i.d. random variables behave asymptotically normal with the usual square root scaling just like the geometric means of the given random variables. Somewhat surprisingly, the multiplicative Cauchy quotient means of i.i.d. random variables behave asymptotically in a rather different way: in order to get a non-trivial normal limit distribution a time dependent centering is needed.

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