A lemma on smoothing (Q578726)

Generalizing results of \textit{T. J. Sweeting} [Ann. Probab. 5, 28-41 (1977; Zbl 0362.60041)] and \textit{L. V. Osipov} and \textit{V. I. Rotar}' [Theor. Veroyatn. Primen. 29, No.2, 366-373 (1984; Zbl 0544.60015); English translation in Theory Probab. Appl. 29, 375-383 (1985)] the author proves the following lemma: Let P, Q, K be probability measures on \({\mathbb{R}}^ k\) and let \(D=P-Q\), \(K_{\epsilon}(A)=K(\epsilon^{-1}A)\) where \(\epsilon >0\) and \(A\subset {\mathbb{R}}^ k\). Let \(B\subset {\mathbb{R}}^ k\) be a balanced set such that \(\alpha =K(B)>1/2\). For \(\epsilon,\epsilon',r,t>0\) denote \[ \gamma =\sup \max_{x\in t\epsilon 'B}\{| D*K_{\epsilon}(A+\epsilon B+x)|,\quad | D*K_{\epsilon}(A\setminus (A^ c-\epsilon B)+x)| \}, \] \[ \tau =\sup_{x\in t\epsilon 'B}Q((A+2\epsilon B)\cap (A^ c-2\epsilon B)+x),\quad \zeta (r)=K(rB^ c). \] Then for all \(t>0\) and \(\epsilon' > \epsilon > 0\) we have \[ | D(A)| \leq (2\alpha -1)^{-1} (\gamma + z + \zeta(\epsilon'\epsilon^{-1})) + ((1- \alpha)/\alpha)^{t-1}. \]