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The fixed points of the multivariate smoothing transform

Abstract

Let \((\mathbf {T}_1, \mathbf {T}_2, \ldots )\) be a sequence of random \(d\times d\) matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors \(X\) with nonnegative entries, satisfying

$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$
(*)

where \(\mathop {=}\limits ^{{\mathcal {L}}}\) denotes equality of the corresponding laws, \((X_i)_{i \ge 1}\) are i.i.d. copies of \(X\) and independent of \((Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )\). For \(d=1\), this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions \(X\) to (*) in the non-critical case, and existence results in the critical case.

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Abbreviations

\(\mathbf {1}, \tilde{\mathbf {1}}\) :

\(\mathbf {1}=(1, \ldots ,1)^\top \in {{\mathbb {R}}^d_\ge }\), \(\tilde{\mathbf {1}}=d^{-1/2} \mathbf {1}\in {\mathbb {S}}_\ge \)

\(\mathop {=}\limits ^{{\mathcal {L}}}\) :

Same law

\(\mathrm {int}\left( A\right) \) :

Topological interior of the set \(A\)

\(\left[ \cdot \right] _{v}\) :

Shift operator in \(\mathbb {V}\), see (4.1)

\(\alpha \)-Regular:

See Definition 2.1.

\({\mathfrak {B}}_n\) :

Filtration, \({\mathfrak {B}}_n=\sigma \Bigl ((T(v))_{\left| {v} \right| <n} \Bigr )\)

\( {\mathfrak {B}}_{{\mathcal {I}}_{t}^u} \) :

\({\mathfrak {B}}_{{\mathcal {I}}_{t}^u} := \sigma \left( U(\emptyset ), \{ T(v) \ : \ v \text { has no ancestor in } {\mathcal {I}}_{t}^u \}\right) \)

\((C)\) :

Condition imposed on the supp of \(\mu \), see Definition 1.1

\(D, {D_{L,s}}\) :

\(D(x) := \frac{1-{\phi }(x)}{H^\alpha (x)}\), \({D_{L,s}}(u,t) = \frac{1- {\phi }(e^{s+t}u)}{H^\alpha (e^tu) e^{\alpha s}L(e^s)}\)

\( \mathbb {E}^\alpha _u\) :

Expectation symbol of \(\mathbb {P}^\alpha _u\), see (3.4).

\( \mathcal {E}_{\alpha ,c}\) :

Extremal points of \(H_{\alpha ,c}^K\), see Lemma 10.5

\(H^s\) :

\(H^s(x) = \int _{{\mathbb {S}}_\ge } \left\langle x,y \right\rangle ^s \, \nu ^{s}(dy)\) for all \(x \in {{\mathbb {R}}^d_\ge }\). It is a \(s\)-homogeneous function, i.e. \(H^s(x)=\left| {x} \right| ^s H^s(x/\left| {x} \right| )\) and satisfies \((P_*^{s}H^s)(u)=k(s) H^s(u)\), \(u \in {\mathbb {S}}_\ge \)

\(I_\mu \) :

\(I_\mu = \{ s \ge 0 \ : \ \mathbb {E}\left\| {\mathbf {M}} \right\| ^s < \infty \}\)

\(\iota (\mathbf{a})\) :

\(\iota (\mathbf{a}) := \inf \{ x \in {\mathbb {S}}_\ge \ : \ \left| {\mathbf{a}x} \right| \}\)

\( {\mathcal {I}}_{t}^u \) :

Stopping line, \({\mathcal {I}}_{t}^u:=\{ v \in \mathbb {V}\ : \ S^u(v)> t \text { and } S^u(v|k) \le t \, \forall \, k<\left| {v} \right| \}\)

\({\mathcal {J}}_\alpha ^K\) :

See Definition 9.4

\(k(s)\) :

Dominant eigenvalue of \(P^{s}\) and \(P_*^{s}\) satisfies \(m(s)=\mathbb {E}N k(s)\)

\(K_C\) :

\(K_C = \left( \min _{y \in C} \min _i y_i \right) \) for compact \(C \subset \mathrm {int}\left( {\mathbb {S}}_\ge \right) \)

\( \mathbf{L}(v)\) :

Recursively defined by \(\mathbf{L}(\emptyset )= \mathbf{Id}\) and \(\mathbf{L}(vi)=\mathbf{L}(v) \mathbf {T}_i(v)\)

\(L\)-\(\alpha \)-elementary:

See Definition 2.1

\(\lambda _\mathbf{a}\) :

Perron–Frobenius eigenvalue of \(\mathbf{a}\in \mathrm {int}\left( {\mathcal {M}}_\ge \right) \)

\(M(u)\) :

Disintegration, see Definition 7.2

\({\mathcal {M}}_\ge \) :

Set \(M(d \times d, {\mathbb {R}}_\ge )\) of nonnegative \(d\times d\) matrices

\(m(s)\) :

\(m(s) = \mathbb {E}N \lim _{n \rightarrow \infty } \left( \mathbb {E}\left\| {\mathbf {M}_{1} \cdots \mathbf {M}_{n}} \right\| ^s \right) ^{1/n}\), where \((\mathbf {M}_{i})_{i \in {\mathbb {N}}}\) are i.i.d. with law \(\mu \)

\( (\mathbf {M}_n)_{n \in {\mathbb {N}}} \) :

Sequence of random matrices (i.i.d. with law \(\mu \) under \(\mathbb {P}\)) c.f. (3.4)

\(\mu \) :

Law on \({\mathcal {M}}_\ge \), defined by \(\int \! f(\mathbf{a}) \mu (d\mathbf{a}) \!:=\! (\mathbb {E}N)^{-1} \, \mathbb {E}\left( \sum _{i=1}^N f(\mathbf {T}_i)\right) \!.\)

\({\mathbb {N}}, {\mathbb {N}}_0\) :

\({\mathbb {N}}= {\mathbb {N}}_0\! \setminus \! \{0\}\).

\(\nu ^{s}\) :

Probability measure on \({\mathbb {S}}_\ge \), satisfy \((P^{s})' \nu ^{s}= k(s) \nu ^{s}\)

\(\mathbb {P}_u\) :

Notation for initial states, \({\mathbb {P}}_{u} \left( {U(\emptyset )=u, U_0=u, S_0=0} \right) =1\)

\(P^{s}\), \(P_*^{s}\) :

Operators on \({\mathcal {C}}^{}\left( {\mathbb {S}}_\ge \right) \) defined in (1.13) resp. (3.1)

\(\varvec{\Pi }_n\) :

\(\varvec{\Pi }_n := \mathbf {M}_n^\top \dots \mathbf {M}_1^\top \)

\( \mathbb {P}^\alpha _u\) :

Exponentially shifted measure, see (3.4).

\({\mathbb {R}}_\ge , {\mathbb {R}}_>\) :

\({\mathbb {R}}_\ge =[0,\infty )\), \({\mathbb {R}}_>=(0,\infty )\)

\(\varrho \) :

Stationary law of \((U(t), R(t))\) under \(\mathbb {P}_u^\alpha \), see Theorem 4.9

\(S^u(v), S_n, S(t)\) :

\(S(v)= - \log \left| {\mathbf{L}(v)^\top u} \right| \), \(S_n := - \log \left| {\varvec{\Pi }_n U_0} \right| \), \(S(t)=S_{\tau _t}\)

\({\mathbb {S}}_\ge \) :

\({\mathbb {S}}_\ge = {\mathbb {S}}\cap {{\mathbb {R}}^d_\ge }\) intersection of the unit sphere and the nonnegative cone in \({\mathbb {R}}^d\)

\(\tau _t\) :

\(\tau _t:= \inf \{n : S_n >t\}\)

\(\mathbb {V}\) :

\(\mathbb {V}=\bigcup _{n=0}^\infty {\mathbb {N}}^n \)

\(U^u(v), U_n, U(t)\) :

\(U(v)= \mathbf{L}(v)^\top \cdot u\), \(U_n= \varvec{\Pi }_n \cdot U_0\), \(U(t)=U_{\tau _t}\)

\(v_\mathbf{a}\) :

Perron–Frobenius eigenvalue of \(\mathbf{a}\in \mathrm {int}\left( {\mathcal {M}}_\ge \right) \)

\(W_n, W\) :

Martingale, see (1.14), \(W_n \rightarrow W\), \(W_0(u)=H^{\alpha }(u)\)

\(W^*\) :

Particular fixed point of \({\mathcal {S}}_Q\), see Lemma 5.2

\(W_{{\mathcal {I}}_{t}^u}^f\) :

\( W_{{\mathcal {I}}_{t}^u}^f = \sum _{v \in {\mathcal {I}}_{t}^u} H^\alpha (\mathbf{L}(v)^\top U(v)) \, f(U(v), S(v)-t),\)

\(Z(u)\) :

\(Z(u):=- \log M(u)\)

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Acknowledgments

The major part of this work was done while the author held a position at the Institute of Mathematical Statistics, University of Münster, which was supported by Deutsche Forschungsgemeinschaft via SFB 878. An earlier version of a part of these results has been published within the author’s Ph.D. thesis [60], written under the supervision of Gerold Alsmeyer, Münster, to whom I’d like to express my gratitude; as well as to Dariusz Buraczewski, Ewa Damek, Yves Guivarc’h, Konrad Kolesko, Daniel Matthes and Matthias Meiners for very helpful discussions on the subject. The author is grateful to the referees for a very careful reading of the paper and many helpful comments, that led to an improvement of the presentation.

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Correspondence to Sebastian Mentemeier.

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Research supported by Deutsche Forschungsgemeinschaft (SFB 878).

Appendix A: Inequalities for Laplace transforms

Appendix A: Inequalities for Laplace transforms

If \({\phi }\) is the LT of a r.v. \({\mathbb {R}}_\ge \), then \(t^{-1}(1-{\phi }(t))\) is again a LT of a measure on \({\mathbb {R}}_\ge \) (see Feller [28], XIII (2.7)]). Consequently, it is decreasing and thus for all \(t \in {\mathbb {R}}_\ge \), \(0 < a < 1\):

$$\begin{aligned} \frac{1-{\phi }(at)}{at} \ge \frac{1-{\phi }(t)}{t} \quad \Rightarrow 1-{\phi }(at) \ge a (1-{\phi }(t)), \end{aligned}$$

as well as, for \(b\ge 1\),

$$\begin{aligned} 1- {\phi }(bt) \le b (1-{\phi }(t)). \end{aligned}$$

This proves the first four inequalities in the subsequent lemma:

Lemma 15.1

Let \({\phi }\) be the Laplace transform of a distribution on \({{\mathbb {R}}^d_\ge }\), \(u \in {\mathbb {S}}_\ge \), \(t \in {\mathbb {R}}_\ge \) and \(\mathbf{A} \in M(d \times d, {\mathbb {R}}_\ge )\). Then

$$\begin{aligned} 1 - {\phi }(atu)&\le 1- {\phi }(tu) \quad \text { for } a < 1, \end{aligned}$$
(15.1)
$$\begin{aligned} 1-{\phi }(atu)&\ge a(1-{\phi }(tu)) \quad \text { for } a < 1, \end{aligned}$$
(15.2)
$$\begin{aligned} 1- {\phi }(btu)&\ge 1- {\phi }(tu) \quad \text { for } b >1, \end{aligned}$$
(15.3)
$$\begin{aligned} 1- {\phi }(btu)&\le b(1-{\phi }(tu)) \quad \text { for } b > 1. \end{aligned}$$
(15.4)
$$\begin{aligned} 1- {\phi }(tu)&\le 1- {\phi }(t\mathbf {1}) \end{aligned}$$
(15.5)
$$\begin{aligned} 1- {\phi }(t\mathbf{A}u)&\le 1- {\phi }(t\left| {\mathbf{A}u} \right| \mathbf {1}) \le 1- {\phi }(t\left\| {\mathbf{A}} \right\| \mathbf {1}) \end{aligned}$$
(15.6)
$$\begin{aligned} 1 - {\phi }(t\mathbf{A}u)&\le \left( \left\| {\mathbf{A}} \right\| \vee 1 \right) (1-{\phi }(t\mathbf {1})) \end{aligned}$$
(15.7)
$$\begin{aligned} 1 - {\phi }(tu)&\ge 1 - {\phi }(t(\min _i u_i) \mathbf {1}) \ge (\min _i u_i) (1-{\phi }(t\mathbf {1})) \end{aligned}$$
(15.8)

Proof

Let \(Z\) be a r.v. with LT \({\phi }\). For all \(u \in {\mathbb {S}}_\ge \), \(\left\langle u,Z \right\rangle \le \left\langle \mathbf {1},Z \right\rangle \). Thus

$$\begin{aligned} 1 - {\phi }(tu)&= \mathbb {E}_{} \left( {1-e^{-t \left\langle u,Z \right\rangle }} \right) = \int _0^\infty t e^{-tr} {\mathbb {P}}_{} \left( {\left\langle u,Z \right\rangle > r} \right) dr \\&\le \int _0^\infty t e^{-tr} {\mathbb {P}}_{} \left( {\left\langle \mathbf {1},Z \right\rangle > r} \right) dr = 1- {\phi }(t\mathbf {1}). \end{aligned}$$

From (15.5) and (15.1) now (15.6) follows:

$$\begin{aligned} 1- {\phi }(t\mathbf{A}u)&= 1-{\phi }(t\left| {\mathbf{A}u} \right| \mathbf{A} \cdot u) \mathop {\le }\limits ^{(15.5)} 1-{\phi }(t\left| {\mathbf{A}u} \right| \mathbf {1}) \\&=1-{\phi }(t\frac{\left| {\mathbf{A}u} \right| }{\left\| {\mathbf{A}} \right\| }\left\| {\mathbf{A}} \right\| \mathbf {1}) \mathop {\le }\limits ^{(15.1)} 1- {\phi }(t\left\| {\mathbf{A}} \right\| \mathbf {1}). \end{aligned}$$

Then (15.7) follows by applying (15.1) resp. (15.4) in (15.6).

In order to prove (15.8), observe that

$$\begin{aligned} \left\langle u,Z \right\rangle = \sum _{i=1}^d u_i Z_i \ge \min _i u_i \sum _{i=1}^d Z_i = \min _i u_i \left\langle \mathbf {1},Z \right\rangle . \end{aligned}$$

Then the argument is the same as given for (15.5), with an additional use of (15.2).   \(\square \)

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Mentemeier, S. The fixed points of the multivariate smoothing transform. Probab. Theory Relat. Fields 164, 401–458 (2016). https://doi.org/10.1007/s00440-015-0615-y

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Keywords

  • Smoothing transform
  • Markov random walks
  • General branching processes
  • Multivariate stable laws
  • Multitype branching random walk
  • Choquet–Deny lemma
  • Weighted branching

Mathematics Subject Classification

  • 60G44
  • 60E05
  • 60B20
  • 60B15