Abstract
We consider stochastic equations of the form X = d W1X + W2X′,where (W1, W2), X and X′ are independent, '=d' denotes equality indistribution, EW1 + EW2 = 1 and X =d X′. We discuss existence,uniqueness and stability of the solutions, using contraction arguments andan approach based on moments. The case of {0, 1}-valued W1 and constant W2leads to a characterization of exponential distributions.
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Baringhaus, L., Grübel, R. On a Class of Characterization Problems for Random Convex Combinations. Annals of the Institute of Statistical Mathematics 49, 555–567 (1997). https://doi.org/10.1023/A:1003127114209
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DOI: https://doi.org/10.1023/A:1003127114209