Elementary fixed points of the BRW smoothing transforms with infinite number of summands
The branching random walk smoothing transform \(\mathbb{T}\) is defined as \[ \mathbb{T}: \text{distr}(U_1)\to\text{distr}\left(\sum^L_{i=1}X_iU_i\right), \] where given realizations \(\{X_i\}^L_{i=1}\) of a point process, \(U_1, U_2,\dots\), are conditionally independent identically distributed random variables, and \(0\leq\text{Prob} \{L=\infty\}\leq 1\). Necessary conditions for the existence of any nonnegative fixed points of \(\mathbb{T}\) are obtained. Moreover, there is a uniqueness result which proves that if an elementary fixed point exists, it is unique up to the scale. The notation of an elementary fixed point is introduced by some restrictions on its Laplace-Stieltjes transform. The tail behaviour of some fixed points with finite mean is studied.
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