Ergodic Behaviour of Affine Recursions II

Invariant Measures and Ergodic Theorems

Hans G. Kellerer

Mathematisches Institut der Universität München

Theresienstraße 39, D-8000 München 2, Germany

October 30, 1992

Summary. This paper is concerned with the discrete–time Markov process (X_{n})_{n≥0} solving the stochastic difference equation X_{n} = Y_{n}X_{n−1} + Z_{n} for n ∈ N, where (Y_{n}, Z_{n})_{n∈N} is a sequence of i.i.d. random variables independent of the initial variable X_{0} and, in accordance with most applications, the state space is restricted to R_{+}. In this part the emphasis is on the recurrent case, where existence and uniqueness of an invariant measure as well as mean and pointwise ergodic theorems can be established.

1991 Mathematics Subject Classification. Primary 47A35, 60J05; Secondary 54H20, 60G30.

Introduction

This is the continuation of work begun in … and to be finished in …. It studies affine recursions on R_{+}, i.e. sequences (X_{n})_{n≥0} defined by

X_{n} = Y_{n}X_{n−1} + Z_{n} for n ∈ N.

Here, (Y_{n}, Z_{n})_{n∈N} is a sequence of independent identically distributed R_{+}^{2}– valued random variables which is independent of the initial variable X_{0} ≥ 0. Without loss of generality, the common distribution ν of (X_{n}, Z_{n}), n ∈ N, will be assumed to belong to the class N defined in Section 0.

A central result of Part I then states that the lower and upper limit of (X_{n})_{n≥0} are constants \underline{x} and \overline{x}, independent of the initial law L(X_{o}). Accordingly the sequence is called recurrent in the case \underline{x} < ∞ and transient in the case \underline{x} = ∞. The aim of Part II is a more specific study of the asymptotic behaviour in the recurrent case; its contents are summarized below.

Section 4. For information about limit points of the sequence (X_{n})_{n≥0} different from \underline{x} and \overline{x} existence and uniqueness of invariant measures for the corresponding transition kernel are essential. Here, in accordance with the topological structure of the state space, only locally finite measures are of interest. The support M of such an invariant measure μ satisfies a functional equation (4.1), by which lower and upper limit can be identified as minimum and maximum of M (4.3). Moreover, M inherits connectedness from the support of the joint law L(Y_{n}, Z_{n}) (4.4). An important property of μ itself consists in the fact that the measure μ([0, t]) grows only polynomially for t → ∞, unless the underlying affine maps are expansive almost surely (4.5).

Section 5. The main result of this section is basic for all that follows. To each recurrent sequence (X_{n})_{n≥0} it assigns an essentially unique locally finite invariant measure (5.6). While its existence is settled by usual compactness arguments (5.3), the uniqueness requires an elaborate localization. It is complicated by the fact that the hitting kernel of an interval [0, t] with \underline{x} < t < ∞, though being still stochastic, need no longer be a Feller kernel. To determine the invariant measure explicitly, an integral equation for its Laplace transform is available (5.7), which, however, is more or less of theoretical interest. In the example

(E) X_{n} = Y_{n}X_{n−1} + 1 with P(Y_{n} = \frac{1}{2}) = \frac{1}{2} = P(Y_{n} = 2)

it leads to the functional equation

ψ(u) = e^{−u} \frac{1}{2} (ψ(\frac{u}{2}) + ψ(2u)) for u > 0,

which can hardly be solved.

Section 6. In view of the computational problems it is important to study the invariant measure μ at least qualitatively. The first assertion concerns its support M, which, whenever unbounded, can be shown to be an interval (6.1). More precisely, here as in some of the following results, P(Y_{n} = 0) = 0 has to be assumed. It is not hard to show μ to be either absolutely continuous or singular with respect to Lebesgue measure (6.2). The first statement applies, unless the joint law L(Y_{n}, Z_{n}) is singular (6.3). It is more involved to prove that μ, apart from a trivial exception, is nonatomic (6.4). The section closes with a stability result that may be used for approximating the invariant measure (6.5).

Section 7. Here the main ergodic theorems for ratios are established. While the version for means follows easily from earlier results (7.1), for the pointwise version a localization as in Section 5 is necessary (7.4). It makes essential use of the fact that the hitting kernel from Section 5 enjoys at least some weaker Feller property. As a consequence of the ergodic theorems the support M and its complement can be identified with the conservative and dissipative part of the process (7.5). If specialized to the example (E) above, this means that the set of limit points of the sequence (X_{n})_{n≥0} equals the interval [2, ∞] almost surely. Rather general results on irreducibility and aperiodicity conclude the section (7.6).

4. Excessive and invariant measures

Basic for the ergodic theorems in Sections 5 and 7 are some properties of measures μ ∈ M(R_{+}) that are excessive or invariant with respect to ν. Here the reference to ν, suppressed in general, actually refers to the corresponding kernel P. Moreover, it should be noted that for this section it is irrelevant whether the underlying distribution ν is recurrent or transient. The results concern mainly the support of μ, where in the first assertion \overline{A} denotes the closure of a subset A of R_{+}:

(4.1) Proposition. With the mapping

h : R_{+} × R_{+}^{2} ∋ (x; y, z) → xy + z ∈ R_{+}

the support M of a measure μ ∈ M(R_{+}) satisfies:

(a) M ⊃ \overline{h[M × N]}, if μ is excessive,

(b) M = \overline{h[M × N]}, if μ is invariant.

Proof. The measure μP is the image of μ ⊗ ν under the continuous mapping h. Therefore, due to a general result from topological measure theory, the support of μP is the closure of the image of

supp (μ ⊗ ν) = M × N

under h, which clearly proves (a) and (b). □

Equally simple is the following auxiliary result:

(4.2) Lemma. If μ ∈ M(R_{+}) is a nontrivial excessive measure with support M, then

\frac{z}{1 − y} ∈ M for (y, z) ∈ N with y < 1.

Proof. With an arbitrary x_{0} ∈ M ≠ ∅ and (y, z) as above, (4.1a) yields recursively

x_{n} := y x_{n−1} + z ∈ M for n ∈ N.

Since M is closed, this implies

\frac{z}{1 − y} = lim_{n→∞} (y^{n}x_{0} + y^{n−1}z + … + z) = lim_{n→∞} x_{n} ∈ M. □

An excessive measure μ obviously preserves this property, if P is replaced by some power P^{n}. Thus it is clear that for (y_{m}, z_{m}) ∈ N with y_{m} < 1 the support M contains not only the fixed points z_{m} / (1 − y_{m}) of the associated affine maps g_{m} : x → y_{m}x+z_{m} but as well those of the composition g_{1} ∘…∘g_{n} for any n ∈ N.

The next result is closely related to (1.3):

(4.3) Theorem. If μ ∈ M(R_{+}) is a nontrivial invariant measure with support M, then, with the notation of (1.3),

(a) inf M = inf\{\frac{z}{1 − y} : (y, z) ∈ N_{c}\},

(b) sup M = sup\{\frac{z}{1 − y} : (y, z) ∈ N_{c}\} if N_{e} = ∅,

= ∞ otherwise.

Proof. 1. If inf M and sup M are denoted by \underline{m} and \overline{m}, respectively, it is immediate from (4.2) that

\underline{m} ≤ inf \{…\} and \overline{m} ≥ sup \{…\}.

Moreover, \overline{m} = ∞ holds in the case N_{e} ≠ ∅. Indeed, (4.1a) implies

(1) M ∩]0, ∞[ ≠ ∅,

and 0 ≠ x_{0} ∈ M combined with (y_{0}, z_{0}) ∈ N_{e} yields, again by (4.1a),

M ∋ y_{0}^{n}x_{0} + y_{0}^{n−1}z_{0} + … + z_{0} → ∞.

2. To prove the inverse inequality for \underline{m}, abbreviate

γ := inf \{\frac{z}{1 − y} : (y, z) ∈ N_{c}\},

which so far may be infinite, and choose δ ∈ [0, γ[. Then

y δ + z ≥ δ for all (y, z) ∈ N,

because this inequality is trivial for y ≥ 1. Therefore

{x : yx + z < δ} ⊂ {x : x < δ} for all (y, z) ∈ N,

while by the invariance of μ on the other hand

∫_{N} μ({x : yx + z < δ}) dν = μ({x : x < δ}) = ∫_{N} μ({x : x < δ}) dν.

Since μ is locally finite, this implies

{x : yx + z < δ} =^{μ} {x : x < δ} for ν-almost all (y, z) ∈ R_{+}^{2}.

If δ varies through [0, γ[, this yields

(yx + z) ∧ γ =^{μ} x ∧ γ for ν-almost all (y, z) ∈ R_{+}^{2}.

Since both sides of this equation are continuous functions of x and (y, z), they agree therefore on the support of μ ⊗ ν, hence

(yx + z) ∧ γ = x ∧ γ for x ∈ M and (y, z) ∈ N.

By choosing (y_{0}, z_{0}) ∈ N with z_{0} > 0 thus

(y_{0}x + z_{0}) ∧ γ = x ∧ γ for all x ∈ M.

This proves indeed the inequality \underline{m} ≥ γ, because any x_{0} ∈ M ∩ [0, γ[ would lead to y_{0}x_{0} + z_{0} = x_{0}, hence y_{0} < 1, and thus to the contradiction

x_{0} = \frac{z_{0}}{1 − y_{0}} ≥ γ.

3. The corresponding proof for \overline{m}, under the additional hypothesis N_{e} = ∅, differs only at the beginning. Define here

γ := sup \{\frac{z}{1 − y} : (y, z) ∈ N_{c}\},

which now may be assumed to be finite, and choose δ ∈ [γ, ∞[. Then

y δ + z ≤ δ for all (y, z) ∈ N,

because N \ N_{c} may contain (y, z) = (1, 0) only. Therefore

{x : yx + z ≤ δ} ⊃ {x : x ≤ δ} for all (y, z) ∈ N,

and the proof continues in complete analogy to part 2. □

While it is easily seen that (b) extends to excessive measures, this fails for (a). Indeed, if ν is transient, the excessive measure ε_{0} (∑_{n≥0} P^{n}) is locally finite according to (2.2b) with inf M = 0, while the infimum on the right–hand side may be arbitrarily large.

A simple consequence of (4.3) is the following result:

(4.4) Proposition. The support M of a nontrivial invariant measure μ ∈ M(R_{+}) is an interval whenever N is connected.

Proof. With 0 ≠ x_{0} ∈ M (see (1) in the proof of (4.3)) the set

M_{n} := {y^{n}x_{0} + y^{n−1}z + … + z : (y, z) ∈ N}

is the continuous image of a connected set, hence

(1) ]\underline{m}_{n}, \overline{m}_{n}[ ⊂ M_{n} ⊂ [\underline{m}_{n}, \overline{m}_{n}] for n ∈ N

with appropriate bounds \underline{m}_{n} and \overline{m}_{n}. Moreover, according to (4.1a) and (4.3)

(2) M_{n} ⊂ M for n ∈ N,

(3) \underline{m}_{n} → inf M and \overline{m}_{n} → sup M,

where in the case N_{e} ≠ ∅ the choice x_{0} ≠ 0 is essential. Together, (1) – (3) prove the assertion. □

As will be seen in (6.1), the converse of (4.4) fails in a surprisingly general sense.

The final part of this section concerns properties not only of the support but of excessive measures themselves. The following technical result will be important:

(4.5) Proposition. Let (y_{0}, z_{0}) ∈ R_{+}^{2} with y_{0} < 1 be given. Then for p > 0 and s > z_{0} / (1 − y_{0}) there exist finite constants

α = α(p) and γ = γ(p, s)

such that under the hypothesis

ν([0, y_{0}[ × [0, z_{0}[) ≥ p

each excessive measure μ ∈ M(R_{+}) satisfies

μ([0, t]) ≤ γ μ([0, s]) t^{α} for t ≥ s.

Proof. If μ is excessive with respect to ν, then

μ([0, t]) ν([0, y_{0}[ × [0, z_{0}[) ≤ (μ ⊗ ν)({(x; y, z) : yx + z ≤ y_{0}t + z_{0}})
≤ μ([0 , y_{0}t + z_{0}]) for all t ≥ 0.

This yields the estimate

μ([0, t]) ≤ p^{−1}μ([0 , y_{0}t + z_{0}])
≤ p^{−2}μ([0 , y_{0}(y_{0}t + z_{0}) + z_{0}]),

hence by iteration

μ([0, t]) ≤ p^{−k}μ([0 , y_{0}^{k} t + y_{0}^{k−1}z_{0} + … + z_{0}])
≤ p^{−k}μ([0 , y_{0}^{k} t + \frac{z_{0}}{1 − y_{0}} ]) for k ∈ N.

Therefore

μ([0, t]) ≤ p^{−k}μ([0, s]) whenever y_{0}^{k} t + \frac{z_{0}}{1 − y_{0}} ≤ s.

This condition holds for k ∈ N satisfying

k ≥ [(log (s − \frac{z_{0}}{1 − y_{0}} ) − log t) / log y_{0}] ≥ 0,

with the convention log y_{0} = −∞ for y_{0} = 0. Since for t ≥ s this bound is nonnegative, finally

μ([0, t]) ≤ p^{−([…] + 1)} μ([0, s]) for t ≥ s,

and a simple computation provides the constants

α := log p / log y_{0} and γ := \frac{1}{p} (s − \frac{z_{0}}{1 − y_{0}} )^{−α}. □

The essential content of this result lies in the fact that the growth of an excessive measure is only polynomial whenever P(Y < 1) > 0.

The last result of this section will be required for stability theorems:

(4.6) Lemma. Let N ∋ ν_{k} →^{w} ν and μ_{k} ∈ M(R_{+}) be excessive with respect to ν_{k}. Then μ_{k} →^{v} μ ∈ M(R_{+}) implies that μ is excessive with respect to ν.

Proof. Vague convergence is implied by weak convergence and compatible with forming product measures, hence

μ_{k} ⊗ ν_{k} →^{v} μ ⊗ ν.

By monotone approximation this yields

∫ g d(μ ⊗ ν) ≤ lim inf_{k→∞} ∫ g d(μ_{k} ⊗ ν_{k}) for 0 ≤ g ∈ C(R_{+}×R_{+}^{2}).

Therefore, with P_{k} denoting the kernel corresponding to ν_{k},

μ P f = ∫ f(yx + z) μ(dx)ν(dy, dz)
≤ lim inf_{k→∞} ∫ f(yx + z) μ_{k}(dx)ν_{k}(dy, dz)
= lim inf_{k→∞} μ_{k}P_{k}f
≤ lim inf_{k→∞} μ_{k}f
= μf for 0 ≤ f ∈ K(R_{+}),

which proves μP ≤ μ. □

It should be mentioned that this result does not carry over to invariant measures, as can be shown by somewhat involved examples.

5. Existence and uniqueness of invariant measures

To derive ergodic theorems in the recurrent case, the following “localization” is essential: (5.1) Definition. Let ν be recurrent and \underline{x} < t < ∞. Then:

(a) ^{t}P denotes the “hitting kernel” belonging to P and [0, t], i.e. tP (x; B) := P^{x}(X_{T} ∈ B) for x ∈ [0, t] and B ∈ B([0, t]),

where

T := inf{n ∈ N : X_{n} ∈ [0, t]};

(b) (^{t}X_{n})_{n≥0} denotes the “sojourn process” belonging to (X_{n})_{n≥0} and [0, t], i.e.

tX_{n} := X_{T_{n}} for n ≥ 0,

where T_{0} < T_{1} < … are the random times when (X_{n})_{n≥0} is in [0, t] and the notation ^{t}X_{n}^{x} is used in the case X_{0} = x.

For easy reference a simple conclusion from probabilistic potential theory is stated explicitly: (5.2) Lemma. Let ν be recurrent and μ ∈ M(R_{+}) be excessive. If ^{t}μ denotes the restriction of μ to [0, t], \underline{x} < t < ∞, then

(a) ^{t}μ is invariant with respect to ^{t}P,

(b) μ is invariant with respect to P. Proof. 1. If I_{A} for A ∈ B(R_{+}) denotes the kernel

I_{A}(x; ·) := 1_{A}(x) ε_{x} for x ∈ R_{+}, the crucial point is the inequality

(1) μ(I_{A} ∑_{n≥0} (P I_{R_{+}\A})^{n}) ≤ μ (see e.g. IX, (62.4) and (31.6), in [8]).

2. Multiplied by P I_{A} from the right and specialized to A = [0, t], (1) yields

(^{t}μ^{t}P)(B) ≤ (μP)(B)
≤ μ(B)
= ^{t}μ(B) for B ∈ B([0, t]).

This proves (a), because ^{t}μ is finite and ^{t}P is a stochastic kernel.

3. For 0 ≤ f ∈ K(R_{+}) choose now t > \underline{x} with supp f ⊂ [0, t] and denote the restriction of f to [0, t] by ^{t}f. Then (a) implies

μf = ^{t}μ^{t}f
= (^{t}μ^{t}P)^{t}f
≤ (μP)f for 0 ≤ f ∈ K(R_{+}),

once more by (1), and this yields the inequality μ ≤ μP needed for (b). □

Now the existence of an invariant measure can be proved in the usual way (see e.g. [14]), under some simplification due to the monotonicity. More generally, the following version is required in Section 7:

(5.3) Proposition. If ν is recurrent and \underline{x} < t < ∞, the measures

ϱ_{n}(B) := ∑_{0≤m<n} P(X_{m} ∈ B) / ∑_{0≤m<n} P(X_{m} ≤ t) for B ∈ B(R_{+}),

defined for n ≥ n_{0} according to (2.2a), satisfy:

(a) {ϱ_{n} : n ≥ n_{0}} is a sequentially compact subset of M(R_{+}),

(b) each limit point μ of the sequence (ϱ_{n})_{n≥n_{0}} is a nontrivial invariant measure.

Proof. (a) For arbitrary s ≥ 0 choose l ∈ N such that

ϑ := P(X_{l}^{s} ≤ t) > 0,

which is possible in view of t > \underline{x}. With μ_{m} := L(X_{m}) this implies by monotonicity

P(X_{m+l} ≤ t) ≥ ∫ _{0≤x≤s} P(X_{l}^{x} ≤ t) μ_{m}(dx)
≥ ∫ _{0≤x≤s} P(X_{l}^{s} ≤ t) μ_{m}(dx)
= ϑ P(X_{m} ≤ s) for m ≥ 0.

With the norming constants

r_{n} := ∑_{0≤m<n} P(X_{m} ≤ t)

this provides the estimate

∑_{0≤m<n} P(X_{m} ≤ s) ≤ ϑ^{−1} ∑_{0≤m<n} P(X_{m+l} ≤ t) ≤ ϑ^{−1} (r_{n} + l).

Since r_{n} → ∞ by (2.2a), this yields

lim sup_{n→∞} ϱ_{n}([0, s]) ≤ ϑ^{−1} < ∞,

i.e. the measures ϱ_{n}, n ≥ n_{0}, are uniformly locally finite, and the assertion follows from general results of topological measure theory.

(b) The assumption ϱ_{n_{k}} →^{v} μ yields μ ≠ 0, because

μ([0, t]) ≥ lim sup_{k→∞} ϱ_{n_{k}}([0, t]) = 1.

With μ_{0} := L(X_{0}), moreover,

ϱ_{n_{k}}f = r_{n_{k}}^{−1} ∑_{0≤m<n_{k}} μ_{0} P^{m}f for 0 ≤ f ∈ K(R_{+}).

By monotone approximation of P f ∈ C(R_{+}) this yields

μ P f ≤ lim inf_{k→∞} ϱ_{n_{k}} P f
= lim inf_{k→∞} r_{n_{k}}^{−1} ∑_{0<m≤n_{k}} μ_{0}P^{m}f
= lim inf_{k→∞} r_{n_{k}}^{−1} ∑_{0≤m<n_{k}} μ_{0}P^{m}f
= μf for 0 ≤ f ∈ K(R_{+}),

because r_{n_{k}} → ∞ and μ_{0}P^{m}f is bounded by max f. Therefore μ is excessive and thus invariant by (5.2b). □

The proof of the uniqueness of μ is much more involved. For the first step (2.6) is crucial:

(5.4) Lemma. Let ν be recurrent and define

C := \{(s, t) : 0 < s < t and \underline{x} < t < ∞\},

\underline{Q}_{s,t}^{x} := lim inf_{n→∞} ∑_{0≤m<n} 1_{[0,s]}(X_{m}^{x}) / ∑_{0≤m<n} 1_{[0,t]}(X_{m}^{x}),

\overline{Q}_{s,t}^{x} := lim sup_{n→∞} ∑_{0≤m<n} 1_{[0,s]}(X_{m}^{x}) / ∑_{0≤m<n} 1_{[0,t]}(X_{m}^{x})

for (s, t) ∈ C and x ∈ [0, t]. Then the set

C^{0} := \{(s, t)∈C: \underline{Q}_{. , .}^{0} and \overline{Q}_{. , .}^{0} are continuous at (s, t) almost surely\},

with λ denoting Lebesgue measure, satisfies

(a) λ^{2}(C \ C^{0}) = 0,

(b) \underline{Q}_{s,t}^{x} = \underline{Q}_{s,t}^{0} a.s. and \overline{Q}_{s,t}^{x} = \overline{Q}_{s,t}^{0} a.s. for x ∈ [0, t]

whenever (s, t) ∈ C^{0}.

Proof. (a) For brevity let Q_{s,t}^{x} stand for \underline{Q}_{s,t}^{x} or \overline{Q}_{s,t}^{x} in the sequel. Then Q_{s,t}^{x} is well–defined, due to X_{0}^{x} ≤ t, and (s, t; ω) → Q_{s,t}^{x}(ω) is a measurable mapping from C × Ω to [0,1]. Therefore

A := \{(s, t; ω) ∈ C × Ω : sup_{k∈N} Q_{s− \frac{1}{k} , t+ \frac{1}{k}}^{0} (ω) < inf_{k∈N} Q_{s+ \frac{1}{k} , t− \frac{1}{k}}^{0} (ω)\}

is a measurable set such that

(1) Q_{. , .}^{0}(ω) is discontinuous at (s, t) if and only if (s, t) ∈ A_{ω}, because Q_{. , .}^{0}(ω) is nondecreasing in s and nonincreasing in t. Again by this monotonicity the section A_{ω} is countable along each line s + t = const., hence

λ^{2}(A_{ω}) = 0 for all ω ∈ Ω.

By Fubini this implies

P(A_{s,t}) = 0 for λ^{2}–almost all (s, t) ∈ C,

and by (1) this verifies assertion (a).

(b) If r stands for s or t, monotonicity in x yields

(2) 1_{[0,r]}(X_{m}^{x}) ≤ 1_{[0,r]}(X_{m}^{0}) for all m ≥ 0.

Now consider δ > 0 small enough to satisfy

δ < s ∧ (t − s) ∧ (t − \underline{x})

and choose f ∈ K(R_{+}) such that

1_{[0,r−δ]} ≤ f ≤ 1_{[0,r]}.

Then according to (2.6) the set

{m ≥ 0 :1_{[0,r]}(X_{m}^{x}) < 1_{[0,r−δ]}(X_{m}^{0})} ⊂ {m ≥ 0 :f(X_{m}^{x}) = 0, f(X_{m}^{0}) = 1}

is almost surely finite, hence with probability 1

(3) 1_{[0,r]}(X_{m}^{x}) ≥ 1_{[0,r−δ]}(X_{m}^{0}) for almost all m ≥ 0.

Since for r = t the terms on the right–hand side of both (2) and (3) sum up to ∞ almost surely, this implies

Q_{s−δ , t}^{0} ≤ Q_{s,t}^{x} ≤ Q_{s , t−δ}^{0} a.s.

Now assertion (b) follows for δ = 1/k → 0 from the hypothesis (s, t) ∈ C^{0}. □

At the next step the pointwise ergodic theorem enters:

(5.5) Lemma. Let ν be recurrent and μ ∈ M(R_{+}) be a nontrivial invariant measure. Then for (s, t) ∈ C^{0}, with the notation of (5.4):

(a) E(\underline{Q}_{s,t}^{x}) = μ([0, s]) / μ([0, t]) = E(\overline{Q}_{s,t}^{x}) for all x ∈ [0, t],

(b) \frac{1}{n}∑_{0≤m<n} P(^{t}X_{m}^{x} ≤ s) → μ([0, s]) / μ([0, t])

for μ−almost all x ∈ [0, t].

Proof. 1. Since μ([0, t]) is finite and, by (4.3a) and (1.3a), strictly positive, the restriction ^{t}μ of μ to [0, t] may be assumed to be normalized, hence by (5.2a) to be a stationary distribution for ^{t}P. Assume in the sequel that X_{0} is distributed according to (the trivial extension of) ^{t}μ. Then (^{t}X_{n})_{n≥0} is a stationary process, hence the classical ergodic theorem ensures that

(1) lim_{n→∞} \frac{1}{n}∑_{0≤m<n} 1_{[0,s]}(^{t}X_{m}) exists almost surely.

Now the sequence of quotients defining \underline{Q}_{s,t}^{x} and \overline{Q}_{s,t}^{x} in (5.4) arises from the sequence of successive means of (1_{[0,s]}(^{t}X_{n}^{x}))_{n≥0} through “extension to the right by constancy” in an evident sense. Therefore it follows from (1) by Fubini that

(2) \frac{1}{n}∑_{0≤m<n} 1_{[0,s]}(^{t}X_{m}^{x}) → Q_{s,t}^{x} a.s. for μ–almost all x ∈ [0, t],

where Q_{s,t}^{x} stands for \underline{Q}_{s,t}^{x} or \overline{Q}_{s,t}^{x}.

2. Next, by stationarity and bounded convergence

μ([0, s]) = (lim_{n→∞}) E(\frac{1}{n}∑_{0≤m<n} 1_{[0,s]}(^{t}X_{m}))
= E(lim_{n→∞} \frac{1}{n}∑_{0≤m<n} 1_{[0,s]}(^{t}X_{m}))
= ∫ _{[0,t]} E(lim_{n→∞} \frac{1}{n}∑_{0≤m<n} 1_{[0,s]}(^{t}X_{m}^{x})) μ(dx)
= ∫ _{[0,t]} E(Q_{s,t}^{x}) μ(dx)

in view of (2). This verifies (a), because E(Q_{s,t}^{x}) by (5.4b) is in fact independent of x ∈ [0, t]. Finally, (a) implies (b) by taking expectations in (2). □

Now one of the main results can be established:

(5.6) Theorem. If ν is recurrent, then there exists a nontrivial invariant measure μ∈M(R_{+}) such that each excessive measure μ′ ∈M(R_{+}) is a multiple of μ.

Proof. Choose μ according to (5.3) and let μ′ ∈ M(R_{+}) be another nontrivial excessive, hence by (5.2b) invariant, measure. For any r choose, moreover, t > r such that

(1) γ := μ([0, t]) > 0 and γ′ := μ′([0, t]) > 0,

(2) λ((C \ C^{0})_{t}) = 0,

which is possible according to (5.4a). If ^{t}μ and ^{t}μ′ denote the restrictions of μ and μ′ to [0, t], the measures γ^{−1} ^{t}μ and (γ′)^{−1} ^{t}μ′ are contained in M_{1}(R_{+}) by (1) and according to (5.5a) agree for sets [0, s] with (s, t) ∈ C^{0}, hence on B([0, t]), by (2). Therefore ^{t}μ and ^{t}μ′ are linearly dependent, which for t → ∞ extends to μ and μ′. □

Two comments on this result are in order:

— Uniqueness holds within the class of locally finite measures only, as is seen already by a deterministic example: If Y = 1/2 = Z, then ν is recurrent with invariant measure μ = ε_{1}. Since x → x/2 + 1/2 is a bijection of the set A := Q ∩]1, ∞[, however, the definition μ′(B) := |A ∩ B| yields another (σ–finite) invariant measure.

— In the transient case nontrivial invariant measures μ ∈ M(R_{+}) may be absent, as follows from (4.3a) in the case Y ≥ 1, or present, as can be shown in the case Y = γ ∈]0, 1[ by a limiting procedure.

It is another question how to get the invariant measure from (5.6). It is not hard to translate the equation μP = μ into an integral equation for the function F (t) := μ([0, t]), but in general it is impossible to solve it. Passing to Laplace transforms simplifies at least the integral equation:

(5.7) Proposition. Let ν and μ be given according to (5.6). Then:

(a) ψ(u) := ∫ e^{−ux}μ(dx) < ∞ for u > 0,

(b) up to a scalar, ψ is uniquely determined by the equation

ψ(u) = ∫ ψ(uy)e^{−uz} ν(dy, dz) for u > 0.

Proof. While (a) is a simple consequence of (4.5), (b) follows from

∫ e^{−ux} (μP)(dx) = ∫ ∫ e^{−u(xy+z)} μ(dx) ν(dy, dz)

and (5.6), because the Laplace transform determines μ. □

If Y and Z are independent (as in the additive or multiplicative model), the integral equation in (5.7b) simplifies to

ψ(u) = E(e^{−uZ}) E(ψ(uY)) for u > 0,

but is still rarely solvable.

6. Main properties of the invariant measure

The measure μ that (5.6) assigns to a recurrent distribution ν actually stands for a one–dimensional family. Nevertheless it will be briefly called “the invariant measure” in this and the following section. As pointed out at the end of the preceding section its quantitative determination is in general out of reach. Thus it is important to obtain at least a qualitative description.

A first property of the support M of μ entered already: (1.3) and (4.3) combine to the equations

inf M = \underline{x} = inf \{\frac{z}{1 − y} : (y, z) ∈ N_{c}\},
sup M = \overline{x} = sup\{\frac{z}{1 − y} : (y, z) ∈ N_{c}\} if N_{e} = ∅,

= ∞ otherwise.

It is a natural question to ask under which conditions, less restrictive than in (4.4), the whole interval [\underline{x}, \overline{x}] is exhausted by M. A surprisingly general answer is given by the following result (for a special case see [2]):

(6.1) Theorem. Let ν be recurrent with P(Y = 0) = 0 and \overline{x} = ∞. Then the invariant measure μ has the support

M = [\underline{x}, ∞[.

Proof. 1. By the hypothesis P(Y =0) = 0 points (0, z)∈N cannot be isolated, hence

\underline{x} = inf \{\frac{z}{1 − y} : (y, z) ∈ N with 0 < y < 1\}.

Thus it suffices to prove that x ∈ M if

x_{0} = \frac{z_{0}}{1 − y_{0}} < x < ∞ with (y_{0}, z_{0}) ∈ N and 0 < y_{0} < 1.

To this end points (y, z) and the associated mappings g : x → yx + z will be identified throughout this proof. It will be accomplished by constructing two sequences n_{1}, n_{2}, … ∈ N and g_{1}, g_{2}, … ∈ N such that

0 = n_{0} ≤ n_{1} ≤ …,
g_{i} = (y_{i}, z_{i}) with y_{i} > 0,
0 ≤ x − x_{k} ≤ (1 − y_{0})^{k}(x − x_{0}) for x_{k} = g_{1} ∘ … ∘ g_{n_{k}} (x_{0}).

Clearly, this implies x ∈ M by (4.1a) and (4.2).

2. It suffices to construct n_{k+1} and g_{n_{k} + 1}, …, g_{n_{k+1}} from n_{k} and g_{1}, …, g_{n_{k}} under the additional assumption x_{k} < x, because otherwise the definition n_{k+1} = n_{k} works. Now the hypothesis \overline{x} = ∞ enters, providing

g′_{i} = (y′_{i}, z′_{i}) ∈ N with y′_{i} > 0 for 1 ≤ i ≤ m

such that

δ_{k} := y_{1} … y_{n_{k}}(g′_{1} ∘ … ∘ g′_{m} (x_{0}) − x_{0}) > x − x_{k}.

Indeed, this follows from lim sup_{n→∞} X_{n}^{x_{0}} = ∞, because the relations Y_{n} > 0 and (Y_{n}, Z_{n}) ∈ N hold almost surely.

3. Let now l ∈ N be defined by

(1) y_{0}^{l−1}δ_{k} > x − x_{k} ≥ y_{0}^{l} δ_{k}.

Then the construction can be continued by n_{k+1} = n_{k} + l + m and

g_{i} := \{ {g_{0}  for n_{k} < i ≤ n_{k} + l, \atop g′_{i−(n_{k} + l)} for n_{k} + l < i ≤ n_{k+1}.}  \}

Indeed, since x_{0} is a fixed point of g_{0} = (y_{0}, z_{0}), by affinity

x_{k+1} − x_{k} = g_{1} ∘ … ∘ g_{n_{k}} ∘ g_{0}^{l} (g′_{1} ∘ … ∘ g′_{m} (x_{0}))
− g_{1} ∘ … ∘ g_{n_{k}} ∘ g_{0}^{l} (x_{0})
= y_{1} … y_{n_{k}}y_{0}^{l} (g′_{1} ∘ … ∘ g′_{m} (x_{0}) − x_{0})
= y_{0}^{l} δ_{k}.

By (1) this implies the required inequalities

x_{k+1} = x_{k} + y_{0}^{l} δ_{k} ≤ x,

x − x_{k+1} = (x − x_{k}) − y_{0} y_{0}^{l−1}δ_{k} < (1 − y_{0})(x − x_{k}). □

Clearly, the condition P(Y = 0) = 0 is essential for this result, as is seen from the trivial case Y = 0, where μ is (a multiple of) the distribution of Z.

The argument used in the following proof can be traced back to Karlin [22]; it applies to locally finite measures as well:

(6.2) Theorem. Let ν be recurrent with P(Y = 0) = 0. Then the invariant measure μ is either absolutely continuous or singular with respect to Lebesgue measure.

Proof. From the equation

(μP)(B) = ∫ _{y>0} μ (\frac{B − z}{y}) dν for B ∈ B(R_{+})

it is clear that μ ≪ λ implies μP ≪ λ as well. If, therefore, μ is decomposed into the absolutely continuous part μ_{c} and the singular part μ_{s}, it follows from

μ_{c}P + μ_{s}P = μP = μ = μ_{c} + μ_{s}

that μ_{c}P ≤ μ_{c}. Thus by (5.6) there exists a constant γ_{c} such that μ_{c} = γ_{c}μ, hence also a constant γ_{s} such that μ_{s} = γ_{s}μ. Now μ_{c}∧μ_{s} = 0 implies γ_{c}∧γ_{s} = 0 and verifies the assertion. □

The remark following (6.1) shows that the condition P(Y = 0) = 0 is essential for (6.2) as well.

It is another question how to decide on the alternative in (6.2). While some special cases are considered in Section 10, the only general result is provided by the following necessary conditions for singularity of μ:

(6.3) Proposition. Let ν be recurrent and the invariant measure μ be singular with respect to Lebesgue measure. Then:

(a) ν is singular with respect to λ^{2},

(b) ν_{y} and ν_{z} are singular with respect to λ, if ν = ν_{y} ⊗ ν_{z}.

Proof. (a) Let B ∈ B(R_{+}) satisfy

λ(B) = 0 and μ(R_{+} \ B) = 0.

Then on the one hand

(1) λ^{2}({(y, z) : yx + z ∈ B}) = ∫ λ(B − yx) λ(dy) = 0 for all x ∈ R_{+},

while on the other hand

0 = μ({x : x ∉ B})
= (μ ⊗ ν)({(x; y, z) : yx + z ∉ B})
= ∫ ν({(y, z) : yx + z ∉ B}) μ(dx).

In view of μ ≠ 0 this ensures the existence of x_{0} ∈ R_{+} such that

(2) ν({(y, z) : yx_{0} + z ∉ B} = 0.

By (1) and (2), therefore, λ^{2} and ν are supported by disjoint sets. (b) With B as above it follows now that

0 = (μ ⊗ ν_{y} ⊗ ν_{z})({(x; y, z) : yx + z ∉ B})
= ∫ ν_{y}({y : yx + z ∉ B}) μ(dx) ν_{z}(dz).

Since supp μ ≠ {0} (see (1) in the proof of (4.3)), this ensures the existence of x_{0} > 0 and z_{0} ≥ 0 such that

λ (\frac{B − z_{0}}{x_{0}} ) = 0 and ν_{y} (R_{+} \ \frac{B − z_{0}}{x_{0}} ) = 0.

The corresponding argument for ν_{z} is even simpler. □

It is an easy consequence of (1.4) that the independence of Y and Z is essential in (b). It applies, however, in the additive or multiplicative model. Moreover, it should be mentioned that (a) and (b) are by no means sufficient for singularity of μ (in this context see Section 10).

The following result is easily established for measures μ containing an atom with maximal mass (see [9]). Since this may fail for locally finite measures, the general proof gets more involved:

(6.4) Theorem. Let ν be recurrent with P(Y = 0) = 0 and \underline{x} < \overline{x}. Then the invariant measure μ is nonatomic.

Proof. 1. It follows as in (6.2) that μ is either nonatomic or purely atomic. It suffices, therefore, to show that in the second case

(1) Z = \underline{x} (1 − Y),

thus in view of (1.4) obtaining a contradiction to the hypothesis \underline{x} < \overline{x}. To this end consider the countable set

R := {x ∈ R_{+} : μ({x}) > 0}.

By the invariance of μ it satisfies

0 = μ(R_{+}\R) = ∑_{x∈R} μ({x}) P (x; R_{+}\R),

hence the definition

P̂(x, x′) := P (x; {x′}) for x, x′ ∈ R

yields a Markov chain on the state space R with invariant measure

μ̂(x) := μ({x}) for x ∈ R.

2. For \underline{x} < t < ∞ define ^{t}P̂ and ^{t}μ̂ in analogy to ^{t}P and ^{t}μ in Section 5. Then ^{t}P̂ is again a stochastic kernel with strictly positive and finite invariant measure ^{t}μ̂. By classical Markov chain theory this implies that all states x ∈ R ∩ [0, t] are (positive) recurrent with respect to ^{t}P̂, hence also recurrent with respect to P̂. For t → ∞ this extends to all states x ∈ R. The recurrent Markov kernel P̂ is in addition irreducible. Indeed, each restriction of μ̂ to a single class yields again an invariant measure for P̂ and its trivial extension to R_{+} an invariant measure for P. Thus by the uniqueness property of μ there can exist only one class. Therefore, again by classical Markov chain theory, every σ–finite (not necessarily locally finite !) excessive measure for P̂ is in fact a multiple of μ̂.

3. Now by the hypothesis P(Y = 0) = 0 the sets

A(x, x′) := {(y, z) : yx + z = x′}, x ∈ R,

are ν–almost disjoint for fixed x′, hence

∑_{x∈R} P̂(x, x′) = ∑_{x∈R} ν(A(x, x′)) ≤ 1 for all x′ ∈ R,

i.e. the equidistribution on R is excessive for P̂. Therefore, according to part 2 of the proof,

(2) μ̂(x) = 1 for all x ∈ R

may be assumed in the sequel.

4. Since μ is locally finite, the support M of μ in view of (2) must consist of isolated points, hence R must contain \underline{x} = min M. Now by (1.2a)

y\underline{x} + z ≥ \underline{x} for ν–almost all (y, z),

which in view of P(Y = 0) = 0 implies

yx + z > \underline{x} for ν–almost all (y, z) whenever x > \underline{x}.

By the invariance of μ̂ this yields

1 = ∑_{x∈R} P̂(x, \underline{x})
= ∑_{x∈R} ν(\{(y, z) : yx + z = \underline{x}\})
= ν(\{(y, z) : y\underline{x} + z = \underline{x}\}),

and (1) is established. □

The remark following (6.1) shows again that the condition P(Y = 0) = 0 is essential for (6.4).

The final result of this section is a strong stability statement, valid under an appropriate normalization:

(6.5) Theorem. Let ν be recurrent with invariant measure μ and assume N ∋ ν_{k} →^{w} ν. If μ_{k} ∈ M(R_{+}) is nontrivial and excessive with respect to ν_{k}, then

(μ_{k}([0, t]))^{−1} μ_{k} →^{v} (μ([0, t]))^{−1} μ whenever t > \underline{x} and μ(\{t\}) = 0.

Proof. By (1.3a) there exists (y_{0}, z_{0}) ∈ N with y_{0} < 1 satisfying the inequality z_{0} / (1 − y_{0}) < t, where actually

ν({(y, z) : y < y_{0}, z < z_{0}}) > 0

may be assumed (slightly increasing y_{0} and z_{0} if necessary). By weak convergence therefore

p := inf_{k∈N} ν_{k}({(y, z) : y < y_{0}, z < z_{0}}) > 0

may be assumed next. Then (4.5) (with the roles of s and t interchanged) provides constants α and γ such that

(1) μ_{k}([0, s]) ≤ γ μ_{k}([0, t]) s^{α} for s ≥ t and all k ∈ N.

Due to μ_{k} ≠ 0 this implies μ_{k}([0, t]) > 0, hence

(2) μ_{k}([0, t]) = 1 for k ∈ N and μ([0, t]) = 1

may be assumed finally. By (1), moreover, the sequence (μ_{k})_{k∈N} is uniformly locally finite, hence (as in the proof of (5.3a)) each subsequence (μ′_{k})_{k∈N} contains a vaguely converging subsubsequence (μ′′_{k})_{k∈N}. By (4.6) and (5.6) its limit is of the form δμ assigning measure 0 to the set {t}. By (2) this implies

δ = δμ([0, t]) = lim_{k→∞} μ′′_{k}([0, t]) = 1,

and the assertion follows. □

This result offers an approximation method for the invariant measure μ associated with a recurrent distribution ν: choose

ν_{k} := \frac{1}{k} ε_{(0,0)} + (1 − \frac{1}{k}) ν for k ∈ N

and observe that an invariant measure μ_{k} for ν_{k} can be determined more or less explicitly (see Section 8).

7. Ratio ergodic theorems

A first information on the fluctuation of a recurrent sequence (X_{n})_{n≥0} by means of its invariant measure is given by the following mean ergodic theorem for ratios, holding without any assumption on the initial law:

(7.1) Theorem. If ν is recurrent with invariant measure μ, then

∑_{0≤m<n} E(f_{1}(X_{m})) / ∑_{0≤m<n} E(f_{2}(X_{m})) → μf_{1} / μf_{2}

for f_{i} ∈ K_{μ}(R_{+}) with μf_{2} ≠ 0.

Proof. Choose t > \underline{x} with μ(\{t\}) = 0 and consider the measures ϱ_{n} defined in (5.3). If (ϱ_{n_{k}})_{k∈N} is any vaguely converging subsequence, (5.3b) and (5.6) imply

(1) ϱ_{n_{k}}f → δμf for all f ∈ K(R_{+}),

where the constant δ satisfies

δμ([0, t]) = lim_{k→∞} ϱ_{n_{k}}([0, t]) = 1,

hence is independent of the subsequence. Since (1) extends from K(R_{+}) to K_{μ}(R_{+}) by monotone approximation, therefore

ϱ_{n}f_{i} → δμf_{i} for i = 1, 2

by (5.3a). Finally, the constant δ disappears by taking quotients. □

The proof of a pointwise analogue of (7.1) is much more involved. The first step consists in proving a counterpart of (5.4):

(7.2) Lemma. Let ν be recurrent with invariant measure μ and for t in

D := ]\underline{x}, ∞[

denote by ^{t}μ the restriction of μ to [0, t]. Then the set

D^{0} := {t ∈ D : P(⋃_{n≥0} {X_{n}^{x} = t}) = 0 for μ−almost all x ∈ [0, t]},

with λ denoting Lebesgue measure, satisfies

(a) λ(D \ D^{0}) = 0,

(b) ^{t}P is a “^{t}μ−F eller kernel” whenever t ∈ D^{0}, i.e.

tP f ∈ C_{^{t}μ}([0, t]) for f ∈ C_{^{t}μ}([0, t]).

Proof. 1. X_{n}^{x}(ω) − t is measurable in (t, x, ω) for all n ∈ N, hence

A := ⋃_{n∈N} {(t, x, ω) : 0 ≤ x ≤ t and X_{n}^{x}(ω) = t}

is a measurable set with

λ(A_{x,ω}) = 0 for all (x, ω),

because the sections A_{x,ω} are countable. Therefore, by Fubini

(μ ⊗ P)(A_{t}) = 0 for λ–almost all t ∈ D. This proves (a), because the last equation is equivalent to

P(A_{t,x}) = 0 for μ–almost all x ∈ [0, t].

2. To verify (b), fix t ∈ D^{0} and f ∈ C_{^{t}μ}([0, t]). Then the set

B_{1} := {x ∈ [0, t] : P(⋃_{n≥0} {X_{n}^{x} = t}) > 0} is a μ–null set, and with the notation

B := {x ∈ [0, t] : f discontinuous at x} this holds as well for the set

B_{2} := {x ∈ [0, t] : ^{t}P (x; B) > 0}. Indeed, ^{t}μ is by (5.2a) invariant with respect to ^{t}P, hence

∫ ^{t}P (x; B)^{t}μ(dx) = ^{t}μ(B) = 0. Since B′ := B_{1} ∪ B_{2} is again a μ–null set, the proof will be completed by establishing

(1) (^{t}P f)(x_{k}) → (^{t}P f)(x_{0}) whenever [0, t] ∋ x_{k} → x_{0} ∉ B′. 3. To this end denote by T_{k} the random time when (X_{n}^{x_{k}})_{n∈N} hits [0, t]

first. Since X_{n}^{x} depends continuously on x,

X_{m}^{x_{0}}(ω) > t for 0 < m < n and X_{n}^{x_{0}}(ω) < t entails

X_{m}^{x_{k}}(ω) > t for 0 < m < n and X_{n}^{x_{k}}(ω) < t

for almost all k ∈ N, hence

{X_{T_{0}}^{x_{0}} ≠ t} ⊂ {T_{k} = T_{0} for almost all k ∈ N}. By assumption x_{0} ∉ B_{1}, hence X_{T_{0}}^{x_{0}} ≠ t with probability 1 and thus

T_{k} → T_{0} a.s., which, once more by the continuity of X_{n}^{x} in x, implies

(2) P(X_{T_{k}}^{x_{k}} → X_{T_{0}}^{x_{0}}) = 1. By assumption x_{0} ∉ B_{2} as well, hence

P(X_{T_{0}}^{x_{0}} ∈ B) = ^{t}P (x_{0}; B) = 0 and thus by the definition of B

(3) P(f continuous at X_{T_{0}}^{x_{0}}) = 1. Since f is bounded, (2) and (3) combine to

E(f(X_{T_{k}}^{x_{k}})) → E(f(X_{T_{0}}^{x_{0}})), and (1) is established. □

At the next step (5.5) is essential: (7.3) Lemma. Let ν be recurrent with invariant measure μ. Suppose, with the notations of (7.2) and (5.4), (∗) t ∈ D^{0} and C_{t}^{0} dense in [0, t] and, in accordance with (4.3), ^{t}μ to be normalized. Then the process (^{t}X_{n})_{n≥0} is stationary and ergodic, if X_{0} is distributed according to (the trivial extension of) ^{t}μ. Proof. 1. The following fact will be used: the measures

ϱ_{n}^{x} := \frac{1}{n}∑_{0≤m<n} ε_{x}(^{t}P)^{m} ∈ M_{1}([0, t])

satisfy

ϱ_{n}^{x} →^{w} ^{t}μ for μ–almost all x ∈ [0, t].

Indeed, this is a consequence of (5.5b), letting s vary through a countable dense subset of C_{t}^{0}. Thus

(1) ϱ_{n}^{x} g → ^{t}μ g for g ∈ C_{^{t}μ}([0, t]) for μ–almost all x ∈ [0, t]. 2. The stationarity of (^{t}X_{n})_{n≥0} is immediate from (5.2a). To prove the

ergodicity it suffices to verify

\frac{1}{n} ∑_{0≤m<n} E(f(^{t}X_{0}, …,^{t}X_{k}) ∏_{1≤i≤l} f_{i}(^{t}X_{m+k+i}))

→ E(f(^{t}X_{0}, …,^{t}X_{k})) E(∏_{1≤i≤l} f_{i}(^{t}X_{i})) for k, l ∈ N and f ∈ C(∏_{0≤i≤k} [0, t]), f_{i} ∈ C([0, t]). With the notations

g(x) := E(∏_{1≤i≤l} f_{i}(^{t}X_{i}^{x})) and h_{n}(x) := ϱ_{n}^{x} g this is equivalent to

(2) E(f(^{t}X_{0}, …,^{t}X_{k}) h_{n}(^{t}X_{k})) → E(f(^{t}X_{0}, …,^{t}X_{k}))^{t}μ g.

3. Now g ∈ C_{^{t}μ}([0, t]) holds even for f_{i} ∈ C_{^{t}μ}([0, t]). Indeed, this is true for l = 1 by (7.2b) and extends to general l by induction, because by the Markov property

g(x) = E(∏_{1≤i<l} f_{i}(^{t}X_{i}^{x}) (^{t}P f_{l})(^{t}X_{l−1}^{x}))

and C_{^{t}μ}([0, t]) is closed with respect to multiplication. Therefore (1) applies and in view of L(^{t}X_{k}) = ^{t}μ yields

h_{n}(^{t}X_{k}) → ^{t}μ g a.s.,

which by bounded convergence verifies (2). □

Now the central result of this paper can be established, holding again without any assumption on the initial law:

(7.4) Theorem. If ν is recurrent with invariant measure μ, then

∑_{0≤m<n} f_{1}(X_{m}) / ∑_{0≤m<n} f_{2}(X_{m}) → μf_{1} / μf_{2} a.s.

for f_{i} ∈ K_{μ}(R_{+}) with μf_{2} ≠ 0.

Proof. 1. It suffices to prove the assertion under the assumption X_{0} = x_{0}, because its general validity then follows by integration. According to (5.4a) and (7.2a) there exists t such that

(1) t > x_{0} and supp f_{i} ⊂ [0, t] for i = 1, 2,

(2) t satisfies condition (∗) in (7.3).

Therefore the classical ergodic theorem, combined with Fubini, yields

(3) \frac{1}{n}∑_{0≤m<n} 1_{[0,s]}(^{t}X_{m}^{x}) → μ([0, s])/μ([0, t]) a.s.

for μ–almost all x ∈ [0, t]. But the sequence defining \underline{Q}_{s,t}^{x} and \overline{Q}_{s,t}^{x} in (5.4) is a trivial extension (as specified in the proof of (5.5)) of the sequence in (3). Thus (5.4b) applies and under the assumption s ∈ C_{t}^{0} implies that (3) in fact holds for all x ∈ [0, t], hence in particular for x_{0}.

2. Now assume f_{i} to be a linear combination of functions 1_{[0,s]}, s ∈ C_{t}^{0}. Then it follows from part 1 that

∑_{0≤m<n} f_{i}(X_{m}) / ∑_{0≤m<n} 1_{[0,t]}(X_{m}) → \frac{1}{μ([0, t])} ∫ _{[0,t]} f_{i} dμ a.s.

Since C_{t}^{0} is dense in [0, t], this convergence extends by monotone approximation first to the case f_{i} ∈ K(R_{+}) and then to the general case. The assertion follows by taking quotients. □

As a first consequence of this pointwise ergodic theorem it can be shown that a recurrent sequence (X_{n})_{n≥0} is in fact recurrent and irreducible in a rather strong sense:

(7.5) Theorem. Let ν be recurrent with invariant measure μ and define the random set

L(ω) := {x ∈ R_{+} : x is limit point of (X_{n}(ω))_{n≥0}}.

Then with probability 1

L(ω) = supp μ.

Proof. 1. Let G_{k}, k ∈ N, be a countable base of R_{+} consisting of bounded sets with a boundary of μ–measure 0 and let t > \underline{x} satisfy μ(\{t\}) = 0. Then by (7.4) with probability 1

∑_{0≤m<n} 1_{G_{k}}(X_{m}) / ∑_{0≤m<n} 1_{[0,t]}(X_{m}) → μ(G_{k}) / μ([0, t])

for all k ∈ N. Therefore

P(∑_{n≥0} 1_{G_{k}}(X_{n}) = ∞ whenever μ(G_{k}) > 0) = 1,

which implies the inclusion

L(ω) ⊃ supp μ a.s.

2. To prove the converse denote by L_{t}(ω) the analogue of L(ω) for the sequence (^{t}X_{n})_{n≥0}. Clearly

L(ω) = ⋃_{\underline{x}<t∈N} L_{t}(ω),

hence it suffices to verify

L_{t}(ω) ⊂ supp μ a.s. for t > \underline{x}.

This is obvious, if X_{0} is distributed as in (7.3), because in this case with probability 1

tX_{n} ∈ supp μ ∩ [0, t] for all n ≥ 0.

Now an application of (2.6) to any function f ∈ K(R_{+}) with f(x) = x on [0, t] shows that the distribution of X_{0} is actually irrelevant. □

Together, (2.2) and (7.5) imply that the two main characterizations of recurrence/transience from classical Markov chain theory carry over to affine recursions in the following form:

— If ν is recurrent, then for x ∈ supp μ always

P^{x}(X_{n} ∈ G infinitely often) = 1,

hence

E^{x}(|{n ≥ 0 : X_{n} ∈ G}|) = ∞,

provided G is an open neighborhood of x.

— If ν is transient, then for x ∈ R_{+} always

E^{x}(|{n ≥ 0 : X_{n} ∈ K}|) < ∞,

hence

P^{x}(X_{n} ∈ K infinitely often) = 0,

provided K is a compact subset of R_{+}.

The final result of this section shows that affine recursions in the recurrent case are not only irreducible but also aperiodic in a strong sense:

(7.6) Proposition. Let ν be recurrent with invariant measure μ. Then for every x_{0} ∈ R_{+} and each open subset G_{0} of R_{+} with μ(G_{0}) > 0 there exists n_{0} ∈ N such that

P^{x_{0}}(X_{n} ∈ G_{0}) > 0 for all n ≥ n_{0}.

Proof. 0. If X_{n} → x a.s., then supp μ = {x} (e.g. by (7.5)). Since the assertion is trivial in this case, \underline{x} < \overline{x} will be assumed henceforth.

1. In a first step m ≥ 0 and s > \underline{x} can be found such that

(1) P(X_{m}^{x} ∈ G_{0}) > 0 for \underline{x} ≤ x < s.

Indeed, P(X\underline{x} ; m ∈ G_{0}) > 0 for some m follows from (7.5) and extends to a neighborhood of \underline{x}, because P^{m} is again a Feller kernel and thus P^{m}1_{G_{0}} is lower semicontinuous.

2. In the next step k ≥ 0 can be found such that

(2) P(\underline{x} ≤ X_{k}^{x_{0}} < s) > 0.

Indeed, (X_{n}^{x_{0}})_{n≥0} hits ]\underline{x}, ∞[ in view of \underline{x} < \overline{x} and stays afterwards in [\underline{x}, ∞[ according to (1.2a), hence visits [\underline{x}, s[ infinitely often by (7.5) – all this with probability 1.

3. In a final step l ≥ 0 can be found such that

(3) P(\underline{x} ≤ X_{l}^{x} < s) > 0 and P(\underline{x} ≤ X_{l+1}^{x} < s) > 0 for \underline{x} ≤ x < s.

Indeed, fix x and choose t > s and l ≥ 0 such that

P(X_{1}^{s} ≤ t) > 0 and P(X_{l}^{t} < s) > 0.

Then by (1.2a) and monotonicity

P(\underline{x} ≤ X_{l}^{x} < s) = P(X_{l}^{x} < s) ≥ P(X_{l}^{t} < s) > 0.

With μ_{1} := L(X_{1}^{x}) moreover

P(\underline{x} ≤ X_{l+1}^{x} < s) ≥ ∫ _{[\underline{x},t]} P(\underline{x} ≤ X_{l}^{x_{1}} < s) μ_{1}(dx_{1}),

where as above

P(\underline{x} ≤ X_{l}^{x_{1}} < s) > 0 for x_{1} ∈ [\underline{x}, t]

and

μ_{1}([\underline{x}, t]) = P(\underline{x} ≤ X_{1}^{x} ≤ t) ≥ P(X_{1}^{s} ≤ t) > 0.

4. By combining k steps according to (2), i times l steps and j times l + 1 steps according to (3), and m steps according to (1), the Markov property yields

P^{x_{0}}(X_{n} ∈ G_{0}) > 0 for n = k + il + j(l + 1) + m.

Since i, j ≥ 0 are arbitrary, the requirement is met by

n_{0} := k + (l − 1)l + m. □

In view of (7.6) the following open problem can be posed: is it possible to strengthen (7.1) to a strong ratio limit theorem as valid for irreducible and aperiodic recurrent random walk?

References

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