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Let ${\tau }_n,n\ge 0$ be a sequence of measure preserving transformations of a probability space (Ω,Σ,P) into itself and let $f_n,n\ge 0$ be a sequence of elements of $L^2(\Omega ,\Sigma ,P)$ with $E{f}_n=0$. It is shown that the distribution of$({\sum }_{i=0}^n{f}_i\circ {\tau }_i\circ ...\circ {\tau }_0){\left(D({\sum }_{i=0}^n{f}_i\circ {\tau }_i\circ ...\circ {\tau }_0)\right)}^{-1}$tends to the normal distribution N(0,1) as n → ∞. 1985 Mathematics Subject Classification: 58F11, 60F05, 28D99.

We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation $V_{[0,x]}(1/|{\tau }^{\text{'}}\left|\right)$ which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.