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The paper is devoted to restricted Oppenheim expansion of real numbers($ROE$),which includes as partial cases already known Engel, Silvester andL\"uroth expansions. We find conditions under which for almost all (withrespect to Lebesgue measure) real numbers from the unit interval their$ROE$-expansion contain arbitrary digit $i$ only finitely many times. Mainresults of the paper states the singularity (w.r.t. Lebesgue measure) of thedistribution of random variable with i.i.d increments of symbols of restrictedOppenheim expansion. General non-i.i.d. case are also studied and sufficientconditions for the singularity of the corresponding probability distributionsare found.

On singularity of distribution of random variables with independent symbols of Oppenheim expansions