Heavy tailed solutions of multivariate smoothing transforms

Abstract

Let $N > 1$ be a fixed integer and $(C_1,..., C_N,Q)$ a random element of $GL(d, \R)^N x \R^d$. We consider solutions of multivariate smoothing transforms, i.e. random variables $R$ satisfying $$R \eqdist \sum_{i=1}^N C_i R_i +Q $$ where $\eqdist$ denotes equality in distribution, and $R, R_1,..., R_N$ are independent identically distributed $\R^d$-valued random variables, and independent of $(C_1,..., C_N, Q)$. We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights $(C_1,..., C_N)$.Comment: 35 page

Power Laws on Weighted Branching Trees

“…In this section we characterize the tail behavior of the distribution of the solution R to the nonhomogeneous equation (1), as defined by (13), when its power-law tail behavior is due to the multiplicative effect of the weights {C i }. The main result is given in the following theorem, which is an application of Theorem 1; see the proof of Theorem 4.1 in [27] and the remark at the end of this subsection.…”

“…. ) be a nonnegative random vector, with N ∈ N ∪ {∞}, P(Q > 0) > 0, and let R be the solution to (1) given by (13). Suppose that there exists j ≥ 1 with P(N ≥ j,C j > 0) > 0 such that the measure P (logC j ∈ du,C j > 0, N ≥ j) is nonarithmetic, and that for some α > 0, 0…”

“…Let us also observe that the solution R, given by (13), to equation (1) may be a constant (non power law) R = r > 0 when P(r = Q + r ∑ N i=1 C i ) = 1. However, similarly as in remark (i), such a solution is excluded from the theorem since…”

“…Hence, such a constructed R is a solution to (22). Also, since R is bounded from above by the solution (13) to the nonhomogeneous linear equation, sufficient conditions for the finiteness of its moments can be obtained from the corresponding results for the solution in (13). Recursion (22) was termed "discounted tree sums" in [1]; for additional details on the existence and uniqueness of its solution see Section 4.4 in [1].…”

“…Such problems are used for modeling memory fragmentation, advance reservation, particle absorption, e.g., see [15] and the references therein. Multidimensional versions of the fixed-point equation (1) have been considered in [39,40] and more recently in [13]. In general, many applied probability problems, appearing in the average case analysis of algorithms and statistical physics, reduce to distributional fixed-point equations of the form…”