Given a sequence $T=(T_i)_{i\geq1}$ of nonnegative random variables, a function f on the positive halfline can be transformed to $\mathbb{E}\prod_{i\geq1}f(tT_i)$. We study the fixed points of this transform within the class of decreasing functions. By exploiting the intimate relationship with general branching processes, a full description of the set of solutions is established without the moment conditions that figure in earlier studies. Since the class of functions under consideration contains all Laplace transforms of probability distributions on $[0,\infty)$, the results provide the full description of the set of solutions to the fixed-point equation of the smoothing transform, $X\stackrel{d}{=}\sum_{i\geq1}T_iX_i$, where $\stackrel{d}{=}$ denotes equality of the corresponding laws, and $X_1,X_2,...$ is a sequence of i.i.d. copies of X independent of T. Further, since left-continuous survival functions are covered as well, the results also apply to the fixed-point equation $X\stackrel{d}{=}\inf\{X_i/T_i:i\geq1,T_i>0\}$. Moreover, we investigate the phenomenon of endogeny in the context of the smoothing transform and, thereby, solve an open problem posed by Aldous and Bandyopadhyay.Comment: Published in at http://dx.doi.org/10.1214/11-AOP670 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Given a sequence (C, T ) = (C, T 1 , T 2 , . . .) of real-valued random variables with T j ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : T j > 0}, the smoothing transform associated with (C, T ), defined on the set P(R) of probability distributions on the real line, maps an element P ∈ P(R) to the law of C + j≥1 T j X j , where X 1 , X 2 , . . . is a sequence of i.i.d. random variables independent of (C, T ) and with distribution P . We study the fixed points of the smoothing transform, that is, the solutions to the stochastic fixed-point equationdrawing on recent work by the authors with J.D. Biggins, a full description of the set of solutions is provided under weak assumptions on the sequence (C, T ). This solves problems posed by Fill and Janson [15] and Aldous and Bandyopadhyay [1]. Our results include precise characterizations of the sets of solutions to large classes of stochastic fixed-point equations that appear in the asymptotic analysis of divide-and-conquer algorithms, for instance the Quicksort equation.

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i \geq 1} T_i X_i$, where $\stackrel{d}{=}$ means equality in distribution, $(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and $X_1,X_2,...$ is a sequence of i.i.d.\ copies of the non-negative random variable $X$ independent of $(C,T_1,T_2,...)$. In this situation, $X$ (or, more precisely, the distribution of $X$) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Further, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C=0. Using this correspondence, we present a full characterization of the set of fixed points under mild assumptions

We consider shot noise processes (X(t)) t≥0 with deterministic response function h and the shots occurring at the renewal epochs 0 = S 0 < S 1 < S 2 . . . of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut)) u≥0 as t → ∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut)) u≥0 converge weakly as t → ∞. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S 1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index −β (with 0 ≤ β < 1/α or 0 ≤ β ≤ α, depending on whether E S 1 < ∞ or E S 1 = ∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut)) u≥0 . The limits are fractionally integrated stable Lévy motions if E S 1 < ∞ and fractionally integrated inverse stable subordinators if E S 1 = ∞.

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