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Absolute continuity of binary geometric sum random variables

Source: Miklós Schweitzer 2017, problem 10

January 13, 2018
probabilityalgebrarandombinary representationcontinuous function

Problem Statement

Let X1,X2,…X_1,X_2,\ldots be independent and identically distributed random variables with distribution P(X1=0)=P(X1=1)=12\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12. Let Y1Y_1, Y2Y_2, Y3Y_3, and Y4Y_4 be independent, identically distributed random variables, where Y1:=∑k=1∞Xk16kY_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}. Decide whether the random variables Y1+2Y2+4Y3+8Y4Y_1+2Y_2+4Y_3+8Y_4 and Y1+4Y3Y_1+4Y_3 are absolutely continuous.
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