Fixed points with finite variance of a smoothing transformation.
The paper deals with fixed points of a transformation of distributions of real random variables with finite variance that maps a distribution \(\mu \) to the distribution of \(\sum _{j=1}^\infty T_jW_j\), where \(W_1,W_2,\dots \) are independent random variables with distribution \(\mu \), and \(T_1,T_2,\dots \) is a sequence of real random variables independent of \(W_1,W_2,\dots \). Within the quite rich literature about distributional fixed points, the paper belongs to the small number of works not requiring the coefficients \(T_1,T_2,\dots \) to be nonnegative, instead it only requires \(E\sum _{j=1}^\infty T_j^2<\infty \) together with some condition assuring \(\sum _{j=1}^\infty T_jW_j\) to converge in \(L_2\). It concentrates on solutions with zero expectation, assuming the convergence condition \(E\sum _{j=1}^\infty | T_j| ^\beta <\infty \) for some \(\beta >2\). For that case, results about the existence and uniqueness of the solution and about the Lebesgue density of its absolutely continuous part are proven, and some investigation of its moment is performed. Moreover, a representation of the characteristic function of the solution by means of fixed points of a distributional transformation with coefficients \(T_1^2,T_2^2,\dots \) is derived, establishing a connection to the more traditionally investigated transformations with nonnegative coefficients. The proofs of the new results rely mainly on the martingale theory and on previous results by \textit{Q. Liu} [Adv. Appl. Probab. 29, 353--373 (1997; Zbl 0901.60055) and ibid. 30, 85--112 (1998; Zbl 0909.60075)], and \textit{J. D. Biggins} [J. Appl. Probab. 14, 630--636 (1977; Zbl 0373.60090)]. The results about the existence and uniqueness are actually formulated in a more general setting, covering also solutions with nonzero expectation, for which the convergence condition \(\sum _{j=1}^\infty T_j\in L_2\) is assumed. However, the proofs for the case of nonzero expectation basically consist in recalling previous results by \textit{R. Lyons} [in: Classical and modern branching processes. IMA Vol Math. Appl. 84, 217--221 (1997; Zbl 0897.60086)] and \textit{U. Rösler} [Stochastic Processes Appl. 42, 195--214 (1992; Zbl 0761.60015)]. The paper is clearly structured and well understandable. A bit confusion cause only some of the interleaving remarks to the presented theorems, due to imprecise formulations and typos. However, the theorems themselves, as well as all other formally stated results, are formulated with full precision, and are rigorously proved.
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