On Berry-Esséen rates, a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean
Let \(X,X_ 1,X_ 2,..\). be a sequence of i.i.d., non-degenerate random variables with mean \(\mu\) and distribution function (df) F. Let \(\theta =F(\mu)\) and for \(n\geq 1\), define \(T_ n\) to be the proportion of \(X_ 1,X_ 2,...,X_ n\) which fall below the sample mean \(\sum^{n}_{k=1}X_ k/n.\) \textit{J. K. Ghosh} [Ann. Math. Stat. 42, 1957-1961 (1971; Zbl 0235.62006)] proved that \(\{T_ n\}\) is asymptotically normal if \(EX^ 2<\infty\) and \(F'(\mu)>0\). It is now shown that this result remains true when \(F'(\mu)=0\); indeed, letting \(\Phi\) denote the standard normal df, \(| P[n^{\frac{1}{2}}(T_ n- \theta)\leq x[\theta(1-\theta)]^{\frac{1}{2}}]-\Phi(x)]\leq cn^{- frac{1}{2}}\) for some \(c>0\) and all x if \(E| X|^ 3<\infty\). Moreover, Ghosh's result is refined by showing that \(\lim \sup_{n\to \infty}n^{\frac{1}{2}}(T_ n-\theta)/(2n \log \log n)^{\frac{1}{2}}=\nu\) almost surely for a specified constant \(\nu\) whenever F''(\(\mu)\) exists. The final result establishes an invariance principle for \(T_ n\) when \(EX^ 4<\infty\).
scientific article; zbMATH DE number 3848311