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Solutions to complex smoothing equations

Abstract

We consider smoothing equations of the form

$$\begin{aligned} X ~\mathop {=}\limits ^{\text {law}}~ \sum _{j \ge 1} T_j X_j + C \end{aligned}$$

where \((C,T_1,T_2,\ldots )\) is a given sequence of random variables and \(X_1,X_2,\ldots \) are independent copies of X and independent of the sequence \((C,T_1,T_2,\ldots )\). The focus is on complex smoothing equations, i.e., the case where the random variables \(X, C,T_1,T_2,\ldots \) are complex-valued, but also more general multivariate smoothing equations are considered, in which the \(T_j\) are similarity matrices. Under mild assumptions on \((C,T_1,T_2,\ldots )\), we describe the laws of all random variables X solving the above smoothing equation. These are the distributions of randomly shifted and stopped Lévy processes satisfying a certain invariance property called \((U,\alpha )\)-stability, which is related to operator (semi)stability. The results are applied to various examples from applied probability and statistical physics.

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Fig. 1
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Notes

  1. 1.

    In slight abuse of language, we will sometimes call a random variable X a solution if the distribution of X is a solution to (1.1).

  2. 2.

    By a \(d \times d\) similarity matrix we mean a \(d \times d\) matrix that can be written as a scale multiple of an orthogonal matrix.

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Acknowledgments

The research has partly been carried out during visits of the authors to the Institute of Mathematical Statistics in Münster. The authors would like to express their gratitude for the hospitality. We further thank Vincent Vargas for interesting discussions on the subject.

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Correspondence to Sebastian Mentemeier.

Additional information

Research supported by short visit Grant 6172 from the European Science Foundation (ESF) for the activity entitled ‘Random Geometry of Large Interacting Systems and Statistical Physics’. Research of M. M. was partially supported by DFG Grant ME 3625/3-1.

Appendices

Appendix 1: The Choquet–Deny lemma

Given a probability measure \(\mu \) on the similarity group \({\mathbb {S}}{} \textit{(d)}\), let U be the closed subgroup generated by the support of \(\mu \)—if \(\mu \) is the step distribution of the associated multiplicative random walk \((L_n)_{n \in {\mathbb {N}}_0}\), see Sect. 3.4, then \(U= {\mathbb {U}}\).

Lemma 5.1

Let \(\psi :U \rightarrow {\mathbb {R}}\) be measurable and bounded. If

$$\begin{aligned} \int _{U} \psi (ug) \, \mu (\mathrm {d} u) = \psi (g) \end{aligned}$$
(5.1)

for all \(u \in U\), then \(\psi \) is constant \(\mu \)-a.e.

This is a consequence of [39, Theorem 3]. For the reader’s convenience, we state that theorem and show how the lemma can be derived from it.

In the following, let G be a locally compact, separable and unimodular group. A probability measure \(\mu \) on G is called aperiodic, if the closed subgroup generated by the support of \(\mu \) equals G. Write [GG] for the commutator subgroup, i.e., the group generated by the commutators \([a,b] :=(ba)^{-1} ab\), \(a,b \in G\), and \(\overline{[G,G]}\) for its closure. Let \(H \subsetneq G\) be a normal subgroup of G. Then G acts on H by conjugation (inner automorphisms), i.e.,

$$\begin{aligned} g.h ~:=~ g^{-1} h g, \quad g \in G, h \in H. \end{aligned}$$

For \(A \subseteq H\) write

$$\begin{aligned} A^G ~:=~ \{g.a:\, g \in G, a \in A\}. \end{aligned}$$

The action of G on H is said to be compact if for each compact \(A \subseteq H {\setminus } \{1_G\}\), \(1_G\) the unit element of G, \(A^G\) is relatively compact, i.e., has compact closure. Then [39, Theorem 3] reads as follows:

Theorem 5.2

Let \(\mu \) be an aperiodic probability measure on G. If \(\overline{[G,G]}\) is Abelian or compact and if the action of G on \(\overline{[G,G]}\) is compact, then the only bounded, measurable functions \(\psi \) satisfying

$$\begin{aligned} \int \psi (u g) \, \mu (\mathrm {d}u) = \psi (g) \quad \text {for all } g \in G \end{aligned}$$

are the \(\mu \)-almost everywhere constant functions.

Following the proof of [25, Theorem A.1], we show how Theorem 5.2 applies to the situation here, i.e., \(G=U\) is the closed subgroup generated by the support of \(\mu \). Then \(\mu \) is aperiodic on U by the very definition of U. Referring to Proposition 4.1, there is a closed subgroup \(\textit{A}_{U}\) of U, which is isomorphic to a closed subgroup of the multiplicative group \({\mathbb {R}}_>\), and a normal compact subgroup \(\textit{C}_{U}=U \cap {\mathbb {O}}{} \textit{(d)}\), such that \(U/\textit{C}_{U}\simeq \textit{A}_{U}\). The groups \(\textit{A}_{U}\) and \(\textit{C}_{U}\) (as a compact group, see [34, Theorem 1.4.1]) are unimodular and hence, by the Fubini formula for the Haar measure on \(U=\textit{A}_{U}\textit{C}_{U}\), [34, Proposition 1.5.5], U is unimodular as well.

Clearly, the commutator subgroup of U is a subgroup of \({\mathbb {O}}{} \textit{(d)}\), hence its closure is compact. Moreover, for any compact \(A \subseteq \overline{[U,U]} {\setminus } \{\textit{I}_{\textit{d}}\}\), \(A^U\) is again a subset of \({\mathbb {O}}{} \textit{(d)}\) since

$$\begin{aligned} u.[a,b] = u^{-1} [a,b]\, u = o^{-1} [a,b]\, o, \end{aligned}$$

where \(u = \left\| u \right\| o\) with \(o \in \textit{C}_{U}\subseteq {\mathbb {O}}{} \textit{(d)}\). Hence, \(A^U\) is relatively compact as a subset of a compact set.

Appendix 2: Evaluating the Lévy integrals

In this section, we compute

$$\begin{aligned} I(x) = \int _{{\mathbb {R}}^d {\setminus } \{0\}} \left( e^{\text {i}\left\langle x,y \right\rangle } - 1 - \text {i}\left\langle x,y \right\rangle \mathbbm {1}_{[0,1]}(\left| y \right| ) \right) \bar{\nu }(\text {d} \textit{y}), \quad x \in {\mathbb {R}}^d \end{aligned}$$
(6.1)

for a deterministic \((U,\alpha )\)-invariant Lévy measure \(\bar{\nu }\), i.e. satisfying (3.29).

Lemma 6.1

Let \(\bar{\nu }\) be a deterministic Lévy measure satisfying (3.29) for some \(1 \ne \alpha \in (0,2)\), and define I(x) via (6.1). Then, for \(0 \not = x \in {\mathbb {R}}^d\),

$$\begin{aligned} I(x) = - |x|^\alpha \eta _1^\alpha (x) + \text {i}|x|^\alpha \eta _2^\alpha (x) + \text {i}\left\langle x,\gamma ^\alpha \right\rangle , \end{aligned}$$
(6.2)

with functions \(\eta _1^\alpha ,\eta _2^\alpha \) defined in (1.19) and (1.20), respectively. \(\eta _1^\alpha \) and \(\eta _2^\alpha \) are bounded real functions satisfying \(\eta _j^\alpha (u^\mathsf {T}x) = \eta _j^\alpha (x)\) for all \(u \in {\mathbb {U}}\), \(x \in {\mathbb {R}}^d{\setminus }\{0\}\), \(j=1,2\), and \(\eta _1^\alpha \) is nonnegative. The vector \(\gamma ^\alpha \) satisfies \(o \gamma ^\alpha = \gamma ^\alpha \) for all \(o \in \textit{C}_{{\mathbb {U}}}\).

Proof

Fix \(0 \not = x \in {\mathbb {R}}^d\) and notice that I(x) is finite since \(\bar{\nu }\) is a Lévy measure. Further, according to Proposition 4.3, there is a \(\textit{C}_{{\mathbb {U}}}\)-invariant finite measure \(\rho \) on \(\textit{S}_{{\mathbb {U}}}\) such that (4.4) holds.

We set

$$\begin{aligned} \gamma ^\alpha :={\left\{ \begin{array}{ll} -\int y \mathbbm {1}_{[0,1]}(|y|) \bar{\nu }(\text {d} \textit{y})&{}\,\\ \quad = - \int _{\textit{A}_{{\mathbb {U}}}} \int _{\textit{S}_{{\mathbb {U}}}} ax \mathbbm {1}_{[0,1]}(|ax|) \left\| a \right\| ^{-\alpha } \rho (\text {d} \textit{x}) \, \textit{H}_{{} \textit{A}_{{\mathbb {U}}}}(\text {d} \textit{a}) &{} \text {if } \alpha < 1 \\ - \int _{\textit{A}_{{\mathbb {U}}}} \int _{\textit{S}_{{\mathbb {U}}}} ax \mathbbm {1}_{(1,\infty )}(|ax|) \left\| a \right\| ^{-\alpha } \rho (\text {d} \textit{x}) \, \textit{H}_{{} \textit{A}_{{\mathbb {U}}}}(\text {d} \textit{a}), &{} \text {if } \alpha \in (1,2). \end{array}\right. } \end{aligned}$$

The asserted \(\textit{C}_{{\mathbb {U}}}\)-invariance follows from the \(\textit{C}_{{\mathbb {U}}}\)-invariance of \(\rho \).

Recalling the definitions

$$\begin{aligned} \eta _1^\alpha (x)&= \frac{1}{|x|^\alpha } \int \big (1 - \cos (\left\langle x,y \right\rangle ) \big ) \nu ^\alpha (\text {d} \textit{y}) \\ \eta _2^\alpha (x)&= \frac{1}{|x|^\alpha } \int \big (\sin (\left\langle x,y \right\rangle ) - \mathbbm {1}_{\{\alpha >1\}}\left\langle x,y \right\rangle \big ) \nu ^\alpha (\text {d} \textit{y}) , \end{aligned}$$

(6.2) holds, and it remains to prove boundedness and invariance properties of \(\eta _i^\alpha \), \(i=1,2\). Let \(u \in {\mathbb {U}}\), then, using (3.29),

$$\begin{aligned} \eta _1^\alpha (u^\mathsf {T}x)&= \frac{1}{\left\| u \right\| ^\alpha |x|^\alpha } \int \big (1 - \cos (\left\langle x,uy \right\rangle ) \big ) \nu ^\alpha (\text {d} \textit{y}) \\&= \frac{\left\| u \right\| ^\alpha }{\left\| u \right\| ^\alpha |x|^\alpha } \int \big (1 - \cos (\left\langle x,y \right\rangle ) \big ) \nu ^\alpha (\text {d} \textit{y}) = \eta _1^\alpha (x), \end{aligned}$$

and the invariance of \(\eta _2^\alpha \) is proved along the same lines. This implies in particular that the continuous functions \(\eta _i^\alpha \) are determined by their respective values on the (relative) compact set \(\textit{S}_{{\mathbb {U}}}\), hence the asserted boundedness follows. \(\square \)

For \(\alpha =1\), we can compute a meaningful expression for \(\eta ^1(x)\) only for \(x \in E_1(Q^\mathsf {T})\). Notice that this implies \(x \in E_1(t^Q)\) for all \(t \ge 0\). Hence, using formula (4.6) for \(\bar{\nu }\), we obtain

$$\begin{aligned} \eta ^1(x)&= \int _{{\mathbb {S}}^{d-1}} \int _{{\mathbb {R}}_>} \bigg ( e^{\text {i}\left\langle (t^{Q})^\mathsf {T}x,s \right\rangle } - 1 - \text {i}\left\langle (t^Q)^\mathsf {T}x,s \right\rangle \mathbbm {1}_{\{|(t^Q)^\mathsf {T}s| \le 1\}} \bigg ) t^{-2}\, \text {d} \textit{t}\, \rho (\text {d} \textit{s}) \nonumber \\&= \int _{{\mathbb {S}}^{d-1}} \int _{{\mathbb {R}}_>} \bigg ( e^{\text {i}\left\langle tx,s \right\rangle } - 1 - \text {i}\left\langle tx,s \right\rangle \mathbbm {1}_{\{t \le 1\}} \bigg ) t^{-2}\, \text {d} \textit{t}\, \rho (\text {d} \textit{s}) \nonumber \\&= -\int _{{\mathbb {S}}^{d-1}} \bigg ( \left| \left\langle x,s \right\rangle \right| + \text {i}\frac{2}{\pi }\left\langle x,s \right\rangle \log |\left\langle x,s \right\rangle | \bigg ) \rho (\text {d} \textit{s}) + \text {i}\left\langle \gamma ,x \right\rangle \end{aligned}$$
(6.3)

for a suitable \(\gamma \in {\mathbb {R}}^d\), see [73, Theorem 14.10] for details.

Appendix 3: Auxiliary results from (Markov) renewal theory

Lemma 7.1

Let \((S_n)_{n \in {\mathbb {N}}_0}\) be a random walk with i.i.d. increments, \(S_0=0\), \({\mathbb {E}}[S_1]>0\) and \({\mathbb {E}}[(S_1^+)^2]<\infty \). Let \(\tau (t) :=\inf \{n \in {\mathbb {N}}_0: S_n > t\}\), \(t \ge 0\). Then, for every \(0<a<1\),

$$\begin{aligned} t {\mathbb {P}}(S_{\tau (t)-1} < at) ~\rightarrow ~ 0 \quad \text {as } t \rightarrow \infty . \end{aligned}$$
(7.1)

Proof

\({\mathbb {E}}[(S_1^+)^2]<\infty \) implies \(\lim _{t \rightarrow \infty } t^2{\mathbb {P}}(S_1 > t) = 0\). Further, it is known from standard random walk theory that \(\lim _{t\rightarrow \infty } t^{-1} {\mathbb {E}}[\tau (t)] = {\mathbb {E}}[S_1]^{-1}\). Consequently, setting, for \(n \in {\mathbb {N}}\), \(A_n :=\{S_0 \le t, \ldots , S_{n-2} \le t, S_{n-1} < at\}\), we have

$$\begin{aligned} t {\mathbb {P}}(S_{\tau (t)-1} < at)= & {} t \sum _{n \ge 1} {\mathbb {P}}(A_n \cap \{S_{n+1}>t\}) \le t \sum _{n \ge 1} {\mathbb {P}}(A_n) {\mathbb {P}}(S_{1}>(1-a)t) \\\le & {} t^2 {\mathbb {P}}(S_{1}>(1-a)t) \cdot \frac{1}{t} \sum _{n \ge 1} {\mathbb {P}}(\tau (t) \ge n) \\= & {} t^2 {\mathbb {P}}(S_1 > (1-a)t) \cdot \frac{{\mathbb {E}}[\tau (t)]}{t} ~\rightarrow ~ 0 \qquad \text {as } t \rightarrow \infty . \end{aligned}$$

\(\square \)

A rate-of-convergence result in Markov renewal theory

Throughout this section, let \(((O_n, S_n))_{n \in {\mathbb {N}}_0}\) be a random walk on \({\mathbb {O}}{} \textit{(d)}\times {\mathbb {R}}\) with increment law \(\mu \), say. The components \(O_n\) and \(S_n\) may be dependent. We assume that \(\mu \) satisfies the minorization condition (M) and that

$$\begin{aligned} {\mathbb {E}}[S_1] > 0 \text { and } {\mathbb {E}}\big [ |S_1|^{ \ell + 1 + \delta } \big ] < \infty \end{aligned}$$
(7.2)

for some \(\ell > 0\). Let \({\mathbb {O}}\) be the closed subgroup of \({\mathbb {O}}{} \textit{(d)}\) generated by the support of \(O_1\).

Proposition 7.2

Let \(\ell >0\) and \(g: {\mathbb {R}}\rightarrow [0,\infty )\) be a measurable function that is decreasing on \([0,\infty )\), with \(g(t)=0\) for all \(t<0\) and \(\lim _{t \rightarrow \infty } t^{ \ell +1+\epsilon } g(t) =0\) for some \(0<\epsilon < (\delta \wedge 1)\). Then

$$\begin{aligned} \lim _{t \rightarrow \infty } \sup _{\left| f \right| \le g} ~t^{ \ell +\epsilon } \left| {\mathbb {E}}\bigg [ \sum _{n=0}^\infty f(O_n, t-S_n) \bigg ] ~-~ \frac{1}{{\mathbb {E}}[S_1]} \int _0^\infty \int _{\mathbb {O}}f(o, r) \, H_{\mathbb {O}}(\text {d} \textit{o}) \, \text {d} \textit{r} \right| = 0. \end{aligned}$$

Here and below, \(\sup _{|f| \le g}\) means the supremum over all measurable functions \(f : {\mathbb {O}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) satisfying \( \sup _{o \in {\mathbb {O}}} |f(o,x)| \le g(x)\) for all \(x \in {\mathbb {R}}\).

The main ingredient in the proof will be the use of regeneration techniques for general state space Markov chains as developed in [10, 68]. We sum up what is needed in the subsequent lemma.

Lemma 7.3

There is a measurable space \((\varOmega , {\mathcal {G}})\) together with a family of probability measures \(({\mathbb {P}}_{o,r})_{o \in {\mathbb {O}}, r \in {\mathbb {R}}}\) and sequences of random variables \(((M_n, R_n))_{n \ge 0}\) and \((\tau _n)_{n \ge 1}\) with

$$\begin{aligned} {\mathbb {P}}_{o,r}\big (((M_n, R_n))_{n \ge 0} \in \cdot \big ) = {\mathbb {P}}\big ( ((oO_n, r + S_n))_{n \ge 0} \in \cdot \big ) \end{aligned}$$
(7.3)

for all \(o \in {\mathbb {O}}\), \(r \in {\mathbb {R}}\). Further, the following properties hold:

  1. (i)

    There is a filtration \(({\mathcal {G}}_n)_{n \in {\mathbb {N}}}\) such that \(((M_n, R_n))_{n \ge 0}\) is Markov adapted to \(({\mathcal {G}}_n)_{n \in {\mathbb {N}}}\), and \((\tau _n)_{n \ge 1}\) is a sequence of predictable \(({\mathcal {G}}_n)_{n \in {\mathbb {N}}}\)-stopping times, i.e., \(\{\tau _n=k\} \in {\mathcal {G}}_{k-1}\).

  2. (ii)

    There are probability measures \(\nu \) on \({\mathbb {O}}\) and \(\eta \) on \({\mathbb {R}}\), \(\eta \) having a bounded Lebesgue density, such that for all \(n \ge 1\) and \(o \in {\mathbb {O}}\), under \({\mathbb {P}}_o\), \(((M_{\tau _n+k}, R_{\tau _n+k}-R_{\tau _n-1}))_{0 \le k \le \tau _{n+1}-\tau _n-1}\) is independent of \((M_0, R_0, \ldots , M_{\tau _n-1}, R_{\tau _n -1})\) and has law \({\mathbb {P}}_{\nu \otimes \eta }(((M_{k}, R_{k}))_{k=0}^{\tau _{1}-1} \in \cdot )\).

  3. (iii)

    For each bounded measurable function \(f:{\mathbb {O}}\rightarrow {\mathbb {R}}\),

    $$\begin{aligned} {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^{\tau _1-1} f(M_n) \bigg ] = {\mathbb {E}}_{\nu \otimes \eta }[\tau _1] \, \int _{\mathbb {O}}f(o) \, H_{\mathbb {O}}(\text {d} \textit{o}). \end{aligned}$$
    (7.4)
  4. (iv)

    There are \(C, \lambda >0\) such that \({\mathbb {P}}_{o,r}(\tau _1 > n) \le C e^{-\lambda n}\) for all \(o \in {\mathbb {O}}\), \(r \in {\mathbb {R}}\) and \(n \in {\mathbb {N}}\).

Here and below, we use the shorthand \({\mathbb {P}}_{\nu \otimes \eta } = \int _{{\mathbb {O}}} \int _{\mathbb {R}}{\mathbb {P}}_{o,r} \, \nu (\text {d} \textit{o}) \, \eta (\text {d} \textit{r})\), the same notation for expectations, and sometimes omit the initial value \(R_0\) if it is irrelevant.

Now we turn to the proof of Proposition 7.2.

Proof (Proof of Proposition 7.2)

In order to prove this result, we will combine methods from [9] and [69]. Below, we describe the steps of the proof and defer the technicalities to several lemmata. By splitting f into its positive and negative part, it suffices to consider nonnegative functions which are bounded by g.

Let \(((M_n, R_n))_{n \ge 0}\) and \((\tau _n)_{n \ge 0}\) be as in Lemma 7.3. Define \(V_0 :=0\) and

$$\begin{aligned} V_n ~:=~ \sum _{k=1}^n (R_{\tau _{k+1}-1}-R_{\tau _k-1}) = R_{\tau _{n+1}-1} - R_{\tau _1-1}. \end{aligned}$$

Then, under each \({\mathbb {P}}_{o,r}\), \((V_n)_{n \ge 1}\) is a random walk with i.i.d. increments and increment law \({\mathbb {P}}_{\nu \otimes \eta }(R_{\tau _1-1} \in \cdot )\) which is absolutely continuous since the law \(\eta \) of \(R_0\) is absolutely continuous. Since \({\mathbb {E}}[S_1]>0\) and

$$\begin{aligned} 1 = {\mathbb {P}}(S_n/n \rightarrow {\mathbb {E}}[S_1] \text { as } n \rightarrow \infty ) = {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(R_n/n \rightarrow {\mathbb {E}}[S_1] \text { as } n \rightarrow \infty ), \end{aligned}$$

we deduce that \({\mathbb {E}}_{\nu \otimes \eta }[V_1] = {\mathbb {E}}_{\nu \otimes \eta }[\tau _1]{\mathbb {E}}[S_1]\).

Let f be nonnegative and set

$$\begin{aligned} \hat{f}(t) ~:=~ {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{k=0}^{\tau _1-1} f(M_k, t- R_k) \bigg ]. \end{aligned}$$

Then we can proceed as in [9, Section 4] to obtain

$$\begin{aligned} {\mathbb {E}}\bigg [&\sum _{n=0}^\infty f(O_n, t-S_n) \bigg ] = {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [ \sum _{n=0}^\infty f(M_n, t-R_n) \bigg ] \nonumber \\&= {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [ \sum _{n=0}^{\tau _1 -1} f(M_n, t-R_n) \bigg ] \nonumber \\&\quad + {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [ \sum _{n=1}^\infty \, {\mathbb {E}}\bigg [\sum _{k=\tau _n}^{\tau _{n+1}-1} f(M_k, t-(R_k-R_{\tau _{n}-1}) - R_{\tau _{n}-1}) \, \Big | \, {\mathcal {G}}_{\tau _n-1} \bigg ] \bigg ] \nonumber \\&= {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [ \sum _{n=0}^{\tau _1 -1} f(M_n, t-R_n) \bigg ] + {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [ \sum _{n=0}^\infty \hat{f}(t-R_{\tau _1-1}-V_n) \bigg ]\nonumber \\&=:E_1(t) + E_2(t). \end{aligned}$$
(7.5)

We show in Lemma 7.4 that \(t^{ \ell +\epsilon } E_1(t)\) tends to zero, uniformly over all f with \(\left| f \right| \le g\). We rewrite

$$\begin{aligned} E_2(t) = \int _{\mathbb {R}}{\mathbb {E}}_{\nu \otimes \eta } \bigg [\sum _{n=0}^\infty \hat{f}(t-s-V_n)\bigg ] \, {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(R_{\tau _1-1} \in \text {d} \textit{s}) \end{aligned}$$

and use (7.4) to infer that (recall that \(f \ge 0\))

$$\begin{aligned} \frac{1}{{\mathbb {E}}_{\nu \otimes \eta } [V_1]} \int _{\mathbb {R}}\hat{f}(r) \, \text {d} \textit{r}&= \frac{1}{{\mathbb {E}}_{\nu \otimes \eta } [V_1]} {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{k=0}^{\tau _1-1} \int _{\mathbb {R}}f(M_k, r-R_k) \, \text {d} \textit{r}\bigg ] \\&= \frac{1}{{\mathbb {E}}_{\nu \otimes \eta } [V_1]} {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{k=0}^{\tau _1-1} \int _{\mathbb {R}}f(M_k, r) \, \text {d} \textit{r}\bigg ] \\&= \frac{1}{{\mathbb {E}}[S_1]} \int _0^\infty \int _{{\mathbb {O}}} f(o,r) \, H_{\mathbb {O}}(\text {d} \textit{o}) \, \text {d} \textit{r}. \end{aligned}$$

Then the claimed convergence rate holds, if

$$\begin{aligned} \int _{\mathbb {R}}\, \sup _{\left| f \right| \le g} t^{ \ell +\epsilon } \left| {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^\infty \hat{f}(t-s-V_n) \bigg ] - \frac{1}{{\mathbb {E}}_{\nu \otimes \eta }[V_1]}\int _0^\infty \hat{f}(r) \, \text {d} \textit{r} \right| \, {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(R_{\tau _1-1} \in \text {d} \textit{s}) \end{aligned}$$

tends to 0 as \(t \rightarrow \infty \). This result will be established in Lemma 7.5. \(\square \)

Lemmata needed in the proof of Proposition 7.2

Proof (Proof of Lemma 7.3)

Observe that \(((oO_n, r+S_n))_{n \in {\mathbb {N}}_0}\) is indeed a Markov chain on \({\mathbb {O}}\times {\mathbb {R}}\) and that the increments of \(S_n-S_{n-1}\) are independent of the past. Since \(\mu \) satisfies the minorization condition (M), we have that for all \(o \in \text {SO}{} \textit{(d)}\), \(B \in \text {SO}{} \textit{(d)}\),

$$\begin{aligned} {\mathbb {P}}(oO_{1} \in B) = {\mathbb {P}}(O_{1} \in o^{-1}B) \ge \gamma H_{{\mathbb {O}}}((o^{-1}B) \cap \text {SO}{} \textit{(d)}) \ge \frac{\gamma }{2} H_{\text {SO}{} \textit{(d)}}(B). \end{aligned}$$
(7.6)

If \({\mathbb {O}}=\text {SO}{} \textit{(d)}\), this shows that \((oO_n)_{n \in {\mathbb {N}}}\) is a Doeblin chain on \({\mathbb {O}}\) (see [66, Section 16.2] for the definition).

If \({\mathbb {O}}\) contains elements with determinant \(-1\) as well, then readily \({\mathbb {O}}={\mathbb {O}}{} \textit{(d)}\), for \(\text {SO}{} \textit{(d)}\subseteq {\mathbb {O}}\), and the product of two matrices with negative determinant has a positive determinant. Moreover, this necessitates \({\mathbb {P}}(\det (O_1)=-1) >0\). Then, for all \(o \in {\mathbb {O}}{\setminus } \text {SO}{} \textit{(d)}\) and \(B \subseteq \text {SO}{} \textit{(d)}\),

$$\begin{aligned} {\mathbb {P}}(oO_{2} \in B) \ge&\int \mathbbm {1}_{\{\det (o')=-1 \}} {\mathbb {P}}(O_{1} \in (oo')^{-1}B) \, {\mathbb {P}}(O_1 \in \text {d} \textit{o}') \nonumber \\ \ge&{\mathbb {P}}(\det (O_1)=-1) \, \frac{\gamma }{2} H_{\text {SO}{} \textit{(d)}}(B). \end{aligned}$$
(7.7)

Thus, \((oO_n)_{n \in {\mathbb {N}}}\) is a Doeblin chain in the case \({\mathbb {O}}={\mathbb {O}}{} \textit{(d)}\), too. Its unique invariant probability measure is given by the normalized Haar measure \(H_{\mathbb {O}}\) on \({\mathbb {O}}{} \textit{(d)}\).

Again by the minorization condition for \(\mu \), it follows that there is an absolutely continuous measure \(\eta (\text {d} \textit{x}):=\mathbbm {1}_I(x) dx\) such that

$$\begin{aligned} {\mathbb {P}}(oO_1 \in A, S_1 \in B) ~\ge ~ \frac{\gamma }{2} H_{\text {SO}{} \textit{(d)}}(A) \eta (B) \end{aligned}$$
(7.8)

for all \(o \in \text {SO}{} \textit{(d)}\) and all measurable AB.

(7.6) and (7.7) yield that there is some \(q <1\) such that, for all \(o \in {\mathbb {O}}\),

$$\begin{aligned} {\mathbb {P}}(oO_k \notin \text {SO}{} \textit{(d)}\text { for } k=1,2) \le q. \end{aligned}$$
(7.9)

It follows that \((oO_n)_{n \in {\mathbb {N}}}\) is \((\text {SO}{} \textit{(d)},\gamma /2,\nu ,1)\)-recurrent in the sense of [9], with \(\nu :=H_{\text {SO}{} \textit{(d)}}\). Then, \(((M_n, R_n))_{n \in {\mathbb {N}}_0}\) can be constructed along the same lines as in [9, Section 3]: Under \({\mathbb {P}}_{o,r}\), let \(((M_n, R_n))_{n \in {\mathbb {N}}}\) have the same transitions as \((oO_n,r+ S_n)_{n \in {\mathbb {N}}}\), but whenever \(O_n\) enters \(\text {SO}{} \textit{(d)}\), an independent \(B(1,\gamma /2)\)-distributed coin is flipped. If 1 shows up, then \((M_{n+1}, R_{n+1}-R_n)\) is generated according to \(\nu \otimes \eta \), this event we call a regeneration; if 0 shows up, then \((M_{n+1}, R_{n+1}-R_n)\) is generated according to \((1-\frac{\gamma }{2})^{-1}\big ({\mathbb {P}}((M_nO_1, S_1) \in \cdot ) - \frac{\gamma }{2} \nu \otimes \eta \big )\). Thus, the total transition probabilities are still equal to that of \((oO_n, r+S_n)_{n \in {\mathbb {N}}}\). Let \(\tau _0=0\) and \(\tau _n\) be the nth regeneration time, see [10, 68] for details. This gives Assertions 1 and 2, while Assertion 3 is proved in [10, Theorem 6.1]. The construction together with (7.9) show that at least every third step, there is a uniform positive chance for regeneration, this yields Assertion 4. \(\square \)

Lemma 7.4

Let g be as in Proposition 7.2 and \(((M_n,R_n))_{n \ge 0}\) as in Lemma 7.3. Then

$$\begin{aligned} \lim _{t \rightarrow \infty } \sup _{\left| f \right| \le g} t^{ \ell +\epsilon +1} \bigg |{\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [\sum _{n=0}^{\tau _1-1} f(M_n, t-R_n) \bigg ] \bigg | = 0. \end{aligned}$$
(7.10)

In particular, \(\lim _{t \rightarrow \infty } t^{ \ell +\epsilon +1} \hat{g}(t)=0\). Moreover,

$$\begin{aligned} \lim _{t \rightarrow \infty } t^{ \ell +\epsilon +1} \, {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(R_{\tau _1-1}>t/2) = 0. \end{aligned}$$
(7.11)

Proof

In order to prove (7.10), we assume that \(g(0) \le 1\) and fix some \(|f| \le g\). Recall that, by Lemma 7.3(iv), \({\mathbb {P}}_{\textit{I}_{\textit{d}},0}(\tau _1 >n) \le C e^{-\lambda n}\) for some \(C,\lambda > 0\). Define \(n_t= (\log t) (\ell +2)/\lambda \), \(t>0\). Then

$$\begin{aligned}&t^{ \ell +\epsilon +1} \bigg | {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [\sum _{n=0}^{\tau _1-1} f(M_n, t-R_n) \bigg ] \bigg | \le t^{ \ell +\epsilon +1} {\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [\sum _{n=0}^{\tau _1-1} g(t-R_n) \bigg ] \\&\quad \le t^{ \ell +\epsilon +1}{\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [\mathbbm {1}_{\{\tau _1 \le n_t\}}\sum _{n=0}^{\lfloor n_t \rfloor } g(t-R_n)\big (\mathbbm {1}_{\{R_n \le t/2\}} + \mathbbm {1}_{\{R_n > t/2\}} \big ) \bigg ] \\&\qquad + t^{ \ell +\epsilon +1}{\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [ \mathbbm {1}_{\{\tau _1 > n_t\}} \sum _{n=0}^{\tau _1-1} g(t-R_n) \bigg ] =:I_1(t) + I_2(t) + I_3(t). \end{aligned}$$

Recall that g is decreasing on \([0, \infty )\) and \(\lim _{t \rightarrow \infty } t^{\ell +1+\delta } g(t)=0\) for some \(\delta >0\). This gives

$$\begin{aligned} I_1(t)= & {} t^{ \ell +\epsilon +1}{\mathbb {E}}_{\textit{I}_{\textit{d}},0} \bigg [\mathbbm {1}_{\{\tau _1 \le n_t\}} \sum _{n=0}^{\lfloor n_t \rfloor } g(t-R_n) \mathbbm {1}_{\{R_n \le t/2\}} \bigg ]\\\le & {} t^{ \ell +\epsilon +1} (\lfloor n_t \rfloor + 1) g(t/2) ~\underset{t \rightarrow \infty }{\rightarrow }~0. \end{aligned}$$

By Jensen’s inequality, \({\mathbb {E}}[\left| S_n \right| ^\kappa ] \le n^{k-1} {\mathbb {E}}[\left| S_1 \right| ^\kappa ]\) for all \(\kappa \ge 1\). Thus, applying Markov’s inequality,

$$\begin{aligned} I_2(t)&\le t^{ \ell +\epsilon +1} \sum _{n=0}^{\lfloor n_t \rfloor } {\mathbb {P}}(S_n > t/2) \le t^{ \ell +\epsilon +1} \sum _{n=0}^{\lfloor n_t \rfloor } \frac{{\mathbb {E}}[ \left| S_n \right| ^{\ell +1+\delta }]}{(t/2)^{\ell +1+\delta }} \\&\le \frac{2^{\ell +1+\delta }}{t^{ \delta -\epsilon }} \sum _{n=0}^{\lfloor n_t \rfloor } n^{\ell +\delta } {\mathbb {E}}[\left| S_1 \right| ^{\ell +1+\delta }] \le 2^{\ell +1+\delta } \frac{n_t^{\ell +1+\delta }}{t^{ \delta -\epsilon }} {\mathbb {E}}[\left| S_1 \right| ^{\ell +1+\delta }], \end{aligned}$$

which tends to zero as \(t \rightarrow \infty \). Finally,

$$\begin{aligned} I_3(t)&\le t^{ \ell +\epsilon +1} {\mathbb {E}}_{\textit{I}_{\textit{d}},0}\big [ \tau _1 \mathbbm {1}_{\{\tau _1 > n_t\}} \big ] \\&\le t^{ \ell +\epsilon +1} \bigg ( n_t {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(\tau _1 > n_t) + \int _{n_t}^\infty {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(\tau _1 > r) \, \text {d} \textit{r}\bigg ) \\&\le t^{ \ell +\epsilon +1} \bigg ( n_t C e^{-\lambda n_t} + \frac{C}{\lambda } e^{-\lambda n_t} \bigg ) \\&\le C \frac{t^{ \ell +\epsilon +1} (n_t +1/\lambda )}{t^{\ell +2}} \rightarrow 0 \end{aligned}$$

as \(t \rightarrow \infty \). Thus we have proved the first and second assertion. (7.11) follows from (7.10) with \(g(s)=\mathbbm {1}_{[0,t/2)}(s)\). \(\square \)

Lemma 7.5

It holds that

$$\begin{aligned} \lim _{t \rightarrow \infty } \int _{\mathbb {R}}\, \sup _{\left| f \right| \le g} t^{ \ell + \epsilon } \left| {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^\infty \hat{f}(t-s-V_n) \bigg ] - \frac{\int _0^\infty \hat{f}(r) \, \text {d} \textit{r}}{{\mathbb {E}}_{\nu \otimes \eta }[V_1]} \right| \, {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(R_{\tau _1-1} \in \text {d} \textit{s}) = 0. \end{aligned}$$
(7.12)

Proof

We start by proving that the functions \(\hat{f}\) are uniformly directly Riemann-integrable over \(\left| f \right| \le g\). Write

$$\begin{aligned} \hat{f}(t) = \int {\mathbb {E}}_{\nu \otimes \delta _0} \bigg [\sum _{k=0}^{\tau _1-1} f(M_k,t-s-R_k) \bigg ] \, \eta (\text {d} \textit{s}), \end{aligned}$$

then observe that

$$\begin{aligned} \int \bigg | {\mathbb {E}}_{\nu \otimes \delta _0} \bigg [\sum _{k=0}^{\tau _1-1} f(M_k,t-R_k) \bigg ] \bigg | \text {d} \textit{t}&\le {\mathbb {E}}_{\nu \otimes \delta _0} \bigg [\sum _{k=0}^{\tau _1-1} \int g(t-R_k) \bigg ] \text {d} \textit{t}\\&= {\mathbb {E}}_{\nu \otimes \delta _0}[\tau _1] \int g(t) \text {d} \textit{t}\end{aligned}$$

and recall that \(\eta \) has a bounded Lebesgue density. Consequently, \(\hat{f}\) is the convolution of a Lebesgue integrable function with a bounded function and hence continuous, see [8, Lemma VII.1.2]. Further, arguing as in [9, Eq. (4.4)],

$$\begin{aligned} \sum _{l\in {\mathbb {Z}}} \sup _{t \in [l h,(l+1)h]} |\hat{f}(t)|&\le {\mathbb {E}}_{\nu \otimes \eta }[\tau _1] \int _{\mathbb {O}}\sum _{l \in {\mathbb {Z}}} \sup _{t \in [l 2h,(l+1)2h]} \left| f(o,t) \right| \, H_{\mathbb {O}}(\text {d} \textit{o}) \\&\le {\mathbb {E}}_{\nu \otimes \eta }[\tau _1] \sum _{l \in {\mathbb {Z}}} \sup _{t \in [l 2h,(l+1)2h]} g(t) =:C_g. \end{aligned}$$

Hence, it suffices to show that g is directly Riemann-integrable. The latter is clear since g is monotone and Lebesgue-integrable (see [8, Proposition V.4.1(v)]). We have the uniform bound (cf. [8, TheoremV.2.4(iii)])

$$\begin{aligned} \sup _{t \in {\mathbb {R}}} \sup _{\left| f \right| \le g} \, {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^\infty |\hat{f}(t-V_n)| \bigg ] \le C_g {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^\infty \mathbbm {1}_{[-4h,4h]}(V_n) \bigg ] =:D_g < \infty . \end{aligned}$$

Now we decompose the integral inside the limit in (7.12) according to the set \(\{R_{\tau _1-1}> t/2\}\) to obtain the following upper bound

$$\begin{aligned} \sup _{s \le t/2} \sup _{\left| f \right| \le g} t^{ \ell + \epsilon } \bigg |{\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^\infty \hat{f}(t-s-V_n) \bigg ] - \frac{1}{{\mathbb {E}}_{\nu \otimes \eta } [V_1]}\int _0^\infty \hat{f}(r) \text {d} \textit{r}\bigg |&\\ +\, 2 D_g \, t^{ \ell + \epsilon } \, {\mathbb {P}}_{\textit{I}_{\textit{d}},0}(R_{\tau _1-1} > t/2)&. \end{aligned}$$

The second term tends to zero by (7.11). For the first term, we invoke [69, Theorem 4.2(ii)] (with \(G=\delta _0\)), which gives (note that \(\left| f \right| \le g\) implies \(|\hat{f}| \le \hat{g}\))

$$\begin{aligned} \lim _{t \rightarrow \infty } t^{ \ell + \epsilon } \sup _{\left| \hat{f} \right| \le \hat{g}} \left| {\mathbb {E}}_{\nu \otimes \eta } \bigg [ \sum _{n=0}^\infty \hat{f}(t-s-V_n) \bigg ] -\frac{1}{{\mathbb {E}}_{\nu \otimes \eta } [V_1]}\int _0^\infty \hat{f}(r) dr \right| = 0 \end{aligned}$$

as soon as \(V_1\) has positive drift (here, \({\mathbb {E}}_{\nu \otimes \eta }[V_1] = {\mathbb {E}}_{\nu \otimes \eta }[\tau _1]{\mathbb {E}}[S_1] > 0\) by the proof of Proposition 7.2), a spread-out law (here, the law of \(V_1\) is even absolutely continuous) and \({\mathbb {E}}_{\nu \otimes \eta }[|V_1|^{ \ell +\epsilon +1}]<\infty \) (which is true by (7.2) and Lemma 7.3(iv)) and \(\hat{g}\) is bounded, Lebesgue-integrable and satisfies

$$\begin{aligned} t^{ \ell + \epsilon } \int _t^{2t} \hat{g}(r) \text {d} \textit{r}\rightarrow 0 \quad \text { and } \quad t^{ \ell + \epsilon } \sup _{r \ge t} \hat{g}(r) \rightarrow 0 \quad \text { as } t\rightarrow \infty . \end{aligned}$$
(7.13)

Lemma 7.4 gives \(\lim _{t \rightarrow \infty } t^{ \ell +\epsilon +1} \hat{g}(t) = 0\), which is sufficient for (7.13) to hold. \(\square \)

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Meiners, M., Mentemeier, S. Solutions to complex smoothing equations. Probab. Theory Relat. Fields 168, 199–268 (2017). https://doi.org/10.1007/s00440-016-0709-1

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Keywords

  • Branching process
  • Characteristic function
  • Infinite divisibility
  • Lévy process
  • Multiplicative martingales
  • Multivariate smoothing equation
  • Similarity matrix

Mathematics Subject Classification

  • 60J80
  • 39B22
  • 60E10