Some Characterizations of Mixed Renewal Processes

The counting process {N_t}_t∈ℝ+ is said to be a mixed renewal process on (Ω, Σ, P) with parameter the d-dimensional (d ∈ ℕ) random vector Θ and interarrival time conditional distribution K(Θ) (or a (P, K(Θ)-MRP for short), if {W_n}_n∈ℕ is P-conditionally independent and

for all n ∈ ℕ.

In particular, for Ψ,Z=ℝ,B and P_Θ((0, ∞)) = 1 a counting process {N_t}_t∈ℝ+ is a P-mixed Poisson process on (Ω, Σ, P) (or a P-MPP for short) with parameter the random variable Θ, if it has P-conditionally stationary independent increments (cf. e.g., [21: Section 4.1, p. 86] for the definition), such that

holds true for each t ∈ (0, ∞).

Note that, for d = 1, P_Θ((0, ∞)) = 1 and K(Θ) = Exp(Θ) P ↾ σ(Θ)-a.s. the (P, K(Θ))-MRP {N_t}_t∈ℝ+ becomes a P-MPP with parameter Θ (see [13: Proposition 4.5]).

1. Let {P_y}_y∈ϒ be an rcp of P over Q, and let f be an inverse-measure-preserving map from Ω into ϒ. Then the following are equivalent:

   (3.1)Pyy∈ϒisconsistentwithf.

   (3.2)PA∩f−1(B)=∫BPy(A)Q( dy)  for each A∈Σ and B∈T.

   (3.3) For each A∈Σ EPχA|σ(f)=P•(A)○f P↾σ(f)-a.s..

2. Let X be a Σ-T-measurable map from Ω into ϒ, let {P_θ}_θ∈Ψ be an rcp of P over P_Θ consistent with Θ, and let k be a Z-T-Markov kernel. If k(·, θ) is the distribution of X under P_θ for θ ∈ Ψ, then the map K(Θ) is a conditional distribution of X given Θ, since by condition (3.3) of (a), we get for A = X⁻¹(B) with B ∈ T that P_X |_Θ(B, ·) = K(Θ)(B, ·) P ↾ σ(Θ)-a.s..

3. Conversely, in the special case where Σ is countably generated and ϒ,T=ℝ,B, given {P_θ}_θ∈Ψ as in (b), we get that for each conditional distribution K(Θ) of X given Θ, there exists an essentially unique probability distribution (P_θ)_X of X, for θ ∈ Ψ, such that for each B∈B, we have

   K(Θ)(B,⋅)=P•X(B)○Θ P↾σ(Θ)-a.s..

In fact, by applying a monotone class argument, it can be easily seen that the rcp is essentially unique in the sense that if Pθ′θ∈Ψ is any other rcp of P over P_Θ which is consistent with Θ, then Pθ=Pθ′ for P_Θ-almost all (P_Θ-a.a. for short) θ ∈ Ψ. But the consistency of {P_θ}_θ∈Ψ together with (a) yields that condition (3.3) holds true; hence setting A = X⁻¹(B) with B∈B, we deduce that k(θ)(B, ·) = (P_•)_X(B) ∘ ΘP ↾ σ(Θ)-a.s..

If no confusion arises, we denote (P_θ)XK(θ) for θ ∈ Ψ.

Throughout what follows, the conditional distributionK(Θ) involving in Remark 1(c) will be considered together with the distributionsK(θ), for θ ∈ Ψ, associated withK(Θ) as in the above remark, without any additional comments.

For the remainder of this section, {P_θ}_θ∈ψis an rcp ofP over P_θ consistent with Θ and {X_i}_i∈Iis a non empty family of Σ-T-measurable maps from Ω into ϒ.

The next result extends Lemma 4.3 from [13].

If {k_i}_i∈Iis a non empty family of Z-T-Markov kernels, then for each i ∈ I and for any fixed B ∈ T the following conditions are equivalent:

1. P_Xi|Θ(B, ·) = K_i(Θ)(B, ·) P ↾ σ(Θ)-a.s.;

2. PθXi−1B=kiB,θforP_Θ-a.aθ ∈ Ψ.

In particular, the same remains true if K_i(Θ)(B, ·) and k_i(B, θ) are independent of i for all B ∈ T and P_θ-a.aθ ∈ Ψ.

Proof. Let us fix on arbitrary i ∈ I. For all B ∈ T and D ∈ Z, we obtain that

Consequently, the equivalence of assertions (i) and (ii) follows. □

Let {k_i}_i∈Ibe as in Lemma 3.1. Suppose that I is countable and T is countably generated. Then the following are equivalent:

1. ConditionP_Xi|ΘK_i(Θ) P ↾ σ(Θ)-a.s. holds true for each i ∈ I;

2. for P_Θ-a.aθ ∈ Ψ-conditionPθ∘Xi−1=ki(⋅,θ)holds true for each i ∈ I.

In particular, the same remains true if K_i(Θ) and k_i(·, θ) are independent of i.

Proof. If (i) holds true, we then get by Lemma 3.1 that for each i ∈ I and B ∈ T condition

is satisfied, which is equivalent to the fact that

where Z₀ := {L ∈ Z : P_Θ(L) = 0}.

Since I is countable and T is countably generated, letting

where GT is a countable generator of T, and applying a monotone class argument, we find a P_Θ-null set L˜I∈Z such that for any θ∉L˜I the equality PθXi−1(B)=ki(B,θ) holds true. So assertion (ii) follows.

Applying a similar reasoning, we obtain the converse implication. □

The following result extends Lemma 4.1 from [13].

Let I be countable and T countably generated. Then the family {X_i} _i∈Iis P-conditionally independent if and only if for P_Θ,-a.a. θ ∈ Ψit isP_θ-independent.

Proof. Assume that {X_i}_i∈I is P-conditionally independent. Then according to Remark 1(a) and following the same reasoning as in the proof of [13: Lemma 4.1], we get that

whenever D ∈ Z, m ∈ ℕ, i₁,…,i_m ∈ I are distinct, and B₁,…, B_m ∈ T, equivalently that for each m ∈ ℕ, for all i₁,…,i_m ∈ I distinct and for all B₁,…, B_m ∈ T there exists a P_Θ-null set L_{I, m, i1,…, im, B1,…, Bm} ∈ Z such that for every θ ∉ L_{I, m, i1,…, im, B1,…, Bm} condition

holds true. Without loss of generality, we may and do assume that m = 2. Since T is countably generated, applying successively two monotone class arguments, we get that there exists a P_Θ-null set L_I ∈ Z such that for any θ ∉ L_I condition (3.4) holds true for m = 2, for each i₁, i₂ ∈ I with i₁ ≠ i₂ and for each B₁, B₂ ∈ T; hence {X_i}_i∈I is P_θ-independent for any θ ∉ L_I. Since the inverse implication is clear, this completes the proof. □

Let I be countable and T countably generated. Then the following hold true:

1. The family {X_i}_i∈Iis P-conditionally identically distributed if and only if for P_θ-a.a. θ ∈ Ψit isP_θ-identically distributed.

2. The family {X_i}_i∈Iis P-conditionally i.i.d. if and only if for P_θ-a.aθ ∈ Ψ it is P_θ-i.i.d..

Proof. Ad (i): If {X_i}_i∈I is P-conditionally identically distributed then for any two i, j ∈ I and for each B ∈ T the equality P_Xi|Θ(B) = P_Xj|Θ(B) holds true P ↾ σ(Θ)-a.s..

Then applying successively Remark 1(a) and a monotone class argument as that in the proof of Lemma 3.2, we find a P_Θ-null set LI′∈Z such that for any θ∉L˜I′ the family {X_i}_i∈I is P_θ-identically distributed. The inverse implication is immediate by Remark 1(a).

Ad (ii): Assume that {X_i}_i∈I is P-conditionally i.i.d.. It then follows by assertion (i) and Lemma 3.3 that there exist two P_Θ-null sets and L˜I′ and L_I in Z such that for any θ∉L^I:=L˜I′∪LI the family {X_i}_i∈I is P_θ-i.i.d.. Since the inverse implication is clear, this completes the proof. □

The counting process {N_t}_t∈ℝ+is a (P, K(Θ))-MRP if and only if for P_θ-a.a. θ ∈ ℝ^dit is a (P_θ, K(θ))-RP.

Proof. Assume that {N_t}_t∈ℝ+ is a (P, K(Θ))-MRP, i.e. that the process {W_n}_n∈ℝ is P - conditionally independent and that for all interarrival times W_n condition P_Wn|Θ = K(Θ) holds true P ↾ σ(Θ)-a.s.. Applying now Lemmas 3.3 and 3.2, we equivalently get that there exist two P_Θ-null sets H_ℕ and H˜ℕ in Z such that for any θ∉H*:=Hℕ∪H˜ℕ the sequence {W_n}_n∈ℕ is P_θ-independent and (P_θ)_wn = K(θ) for each n ∈ ℕ, respectively, i.e. such that {N_t}_t∈ℝ+ is a (P_θ, K(θ))-RP for any θ ∉ H_*. □

Next, we need a preparatory result extending Lemmas 3.3 and 3.4 for uncountable index set. To this aim, we recall some notions more.

A filtration {Ƶ_t}_t∈ℝ+ is said to be right-continuous if Zt=∩s>tZs for any t ∈ ℝ₊. Let I be an arbitrary subset of ℝ₊ and let ϒ be a metric space. We say that the family {X_i;}_i∈I of ∑−Bϒ-measurable maps from Ω into ϒ is separable, if there exists a countable set G ⊆ I such that for each ω ∈ Ω the set {(u, X_u(ω)) : u ∈ G} is dense in {(i, X_i(ω)) : i ∈ I}. Any such set G is called a separator (or separating set) for {X_i}_i∈j.

Let I ⊆ ℝ₊ and let Q be a probability measure on Σ. Then the following can be easily proven:

1. If {U_t}_t∈I is a family of ∑−Bϒ-measurable maps from Ω into ϒ, and {Ƶ_t}_t∈I is its canonical filtration, then {U_t}_t∈I is Q-independent if and only if for every bounded Bϒ-measurable realvalued function f on ϒ the equality

   (3.5)EQχAfUt=Q(A)EQfUt

   holds true for each s, t ∈ I with s < t and for each A ∈ Ƶ_s.

2. If U₁ and U₂ are two ∑−Bϒ-measurable maps from Ω into ϒ, then they are Q-identically distributed if and only if EQfU1=EQfU2 for every bounded Bϒ-measurable real-valued function f on ϒ.

Let ϒ be a Polish space, let {X_t}_t∈ℝ+be a family of∑−Bϒ-measurable maps from Ω into ϒ and letHtt∈ℝ+be its canonical filtration. If the family {X_t}_t∈ℝ+is separable with separator ℚ₊then the following hold true:

1. IfHtt∈ℝ+is right-continuous, then {X_t}_t∈ℝ+ is P-conditionally independent if and only if for P_θ-a.a. θ ∈ Ψ it is P_θ-independent.

2. The family {X_t}_t∈ℝ+ is P-conditionally identically distributed if and only if for P_θ-a.a. θ ∈ Ψ it is P_θ-identically distributed.

3. IfHtt∈ℝ+is right-continuous, then {X_t}_t∈ℝ+is P-conditionally i.i.d. if and only if for P_θ-a.a. θ ∈ Ψ it is P_θ-i.i.d..

1. It follows by Lemma 3.3 that there exists a P_Θ-null set O_ℚ+ ∈ Z such that for any θ ∉ O_ℚ+ condition (3.4) holds true with ℚ₊ and Bϒ in the place of I and T, respectively.

   Throughout this proof fix on an arbitrary θ ∈ O_ℚ+. Then condition (3.4) together with Remark 2(a) implies that for all s, t ∈ ℚ₊ with s < t, for every bounded Bϒ-measurable real-valued function f on ϒ and for each A∈Hs, we have

   (3.6)EPθχAfXt=Pθ(A)EPθfXt.

   If we take s,t ∈ ℝ₊ with s < t and write (3.6) for s′, t′ ∈ ℚ₊ with s′ < t′ and then let s′ ↓ s and t′ ↓ t, the separability of {X_t}_t∈ℝ+ together with an application of Lebesgue’s Dominated Convergence Theorem yields that for all A∈∩s′∈ℚ+,s′>sHs′=Hs and for every bounded continuous real-valued function f on ϒ condition (3.6) holds true.

2. Let s, t ∈ ℝ₊ with s < t and let f be a function as in (3.5). Then for each n ∈ ℕ there exists a bounded continuous real-valued function g_n on ϒ satisfying the inequality ∫gn−fdPθXt≤1n (cf. e.g., [7: Proposition 415P]); hence there exists a sequence {g_n}_n∈ℕ of bounded continuous real-valued functions on ϒ such that condition limn→∞ ∫χAgn−f∘XtdPθ=0 holds true for all A∈Hs, implying together with (a) that EPθχAfXtn→∞=limn→∞ EPθχAgnXt=limn→∞ Pθ(A)EPθgnXt=Pθ(A)EPθfXt for all A∈Hs; hence by Remark 2(a), we get that {X_t}_t∈ℝ+ is P_θ-independent, which proves (i).

   Ad (ii): The “if” implication is immediate by Remark 1(b). For the “only if” part, assume that {X_t}_t∈ℝ+ is P-conditionally identically distributed.

3. Since {X_t}_t∈ℝ+ is P-conditionally identically distributed, we get that for any two s, t ∈ ℚ₊ and for each B∈Bϒ the equality P_Xt|Θ(B) = P_Xs|Θ(B) holds P ↾ σ(Θ)-a.s.. The latter together with Lemma 3.4(i) yields the existence of a P_Θ-null set O˜ℚ+∈Z such that for any θ∉O˜ℚ+ and for all s, t ∈ ℚ₊ condition (P_θ)X_t = (P_θ)_Xs holds true, which by Remark 2(b) equivalently yields that for any θ∉O˜ℚ+, for every function f as in the above remark, and for all s, t ∈ ℚ₊, we have EPθfXt=EPθfXs.

   Till the end of the proof of (ii), fix on an arbitrary θ∉O˜ℚ+.

4. If we take s, t ∈ ℝ₊ and if we write the last equality for s′, t′ ∈ ℚ₊ and then let s′ ↓ s and t′ ↓ t, the separability of {X_t}_t∈ℝ+ together with an application of Lebesgue’s Dominated Convergence Theorem yields that for every bounded continuous real-valued function f on ϒ condition EPθfXt=EPθfXs holds true.

Following now the same reasoning with that of steps (b) and (c), we obtain that the last equality is satisfied by all functions f as in Remark 2, and all s, t ∈ ℝ₊, which is equivalent to the fact that condition (P_θ)_Xt = (P_θ)_Xs holds true for all s, t ∈ ℝ₊; hence (ii) follows.

Ad (iii): Assume that {X_t;}_t∈ℝ+ is P-conditionally i.i.d. and that its canonical filtration is right-continuous. It then follows by assertions (i) and (ii) that there exist two P_Θ-null sets O˜ℚ+ and O_ℚ+ in Z such that for any θ∉O^ℚ+:=O˜ℚ+∪Oℚ+ the family {X_t}_t∈ℝ+ is P_θ-i.i.d.. Since the inverse implication is clear, assertion (iii) follows. □

Let {N_t}_t∈ℝ+be a counting process and letK˜tθt∈ℝ+be a family of probability distributions onBwith parameter θ ∈ Ψ. Then {N_t}_t∈ℝ+hasP-conditionally stationary independent increments such that condition

Proof. Since {N_t}_t∈ℝ+ is a counting process it has right-continuous paths; hence it is separable with separator ℚ₊. Note also that the canonical filtration of {N_t}_t∈ℝ+ is right-continuous (see [20: Theorem 25], where the proof works for any probability space not necessarily complete). Thus, all assumptions of Proposition 3.2 are fulfilled, and so, we may apply it to deduce the thesis of the theorem. □

Corollary 3.3.1([13: Proposition 4.4])

The family {N_t}_t∈ℝ+is a P-MPP with parameter Θ if and only if it is a P_θ-Poisson process with parameter θ for P_θ-a.a. θ∈ ℝ.

LetFbe a σ-subalgebra of Σ and let {X_i}_i∈Ibe a non empty family of Σ-T-measurable maps from Ω into ϒ such that {X_i}_i∈Iis P-conditionally i.i.d. overF. suppose that T is countably generated and that P_χi is perfect for each i ∈ I. Then there exists a probability measure M onT⊗F with marginal R:=P↾Fsuch that M := P ∘ (X_i × id_Ω)⁻¹for every i ∈ I, and a product rcp {Q_ω}_ω∈Ωon T for M with respect to R, such that

1. for any fixedB ∈ T and i ∈ I the map Q_•(B): Ω → [0, 1] isR-a.s. equal toPXi−1B|F⋅;

2. ∫FQωI(H)R( dω)=PF∩X−1(H)for everyF∈FandH ∈ T_i, whereQωIdenotes the I-fold product probability ⊗_i∈IP_i of copies P_i := Q_ω of Q_ω for i ∈ I, and X : Ω → ϒ^I is defined by X(ω) = (X_i(ω))_i∈I for each ω ∈ Ω.

Proof. First fix on an arbitrary i ∈ I.

1. The function X_i × id_Ω from Ω into ϒ × Ω defined by means of

   Xi×idΩ(ω)  :=  Xi(ω),ω  for each ω∈Ω

   is Σ-T ⊗ F -measurable. So, we have a probability measure M_i := P ∘ (X_i × id_Ω)⁻¹ on T⊗F. Since all X_i are P-conditionally identically distributed over F, it follows that M_i is independent of i, so we may write M := M_i* for any fixed i* ∈ I.

2. There exists a product rcp {Q_ω}_ω∈Ω on T for M with respect to R=P↾F such that for any fixed B ∈ T

   Q•(B)=PXi−1(B)∣F(⋅) R− a.s..

   In fact, by assumption each marginal measure P_Xi of M on T is perfect and T is countably generated; hence by [5: Theorem 6], there exists a product rcp {Q_ω}_ω∈Ω on T for M with respect to R.

   Since {Q_ω}_ω∈Ω satisfies (D2), we get that

   ∫FQω(B)R(dω)=M(B×F)=PF∩Xi−1(B)=∫FPXi−1(B)|F(ω)R(dω)

   for every B ∈ T and F∈F, which proves (b); hence (i) follows.

3. Using (i) and a monotone class argument, we get that (ii) holds true. □

   The next result extends a corresponding one due to Olshen (see [18: Theorem (3)]) concerning a generalization of de Finetti’s Theorem.

Let {X_i}_i∈Ibe a P-exchangeable infinite family of Σ-T-measurable maps from Ω into ϒ. suppose that T is countably generated and P_Xi is perfect for each i ∈ I. Then there exists a d-dimensional random vector Θ such that {X_i}_i∈Iis P-conditionally i.i.d. given Θ.

Proof. Since {X_i}_i∈I is P-exchangeable, it follows by [7: Theorem 459B], that there exist a σ-subalgebra F of Σ such that {X_i}_i∈I is P-conditionally i.i.d. over F. So, applying Lemma 4.1, we deduce that there exists a family {Q_ω}_ω∈Ω of F-T-Markov kernels such that

for every H ∈ T₁ and F∈F, where R:=P↾F. Then there exists a countably generated σ-subalgebra A of F such that Q_•(B) is A-measurable for arbitrary but fixed B ∈ T (take, e.g., AB  :=  Q•(B)−1(E):E∈GB for B ∈ T, and A  :=  σ∪B∈GTAB,, where GB and GT is a countable generator of B and T, respectively). Since A is countably generated, there exists a map Θ˜:Ω→ℝ such that A=σΘ˜ (take, e.g., Θ˜ to be the Marczewski functional on Ω, cf. e.g., [6: 343E] for the definition). But since {X_i}_i∈I is P-conditionally i.i.d. over F and A⊆F, it follows that {X_i}_i∈I is so over A=σΘ˜.

Note also that since ℝ and ℝ^d are standard Borel spaces of the same cardinality, there exists a Borel isomorphism g from ℝ into ℝ^d (cf. e.g., [7: Corollary 424D(a)]). So, putting Θ:=g∘Θ˜, we get that Θ is a d-dimensional random vector on Ω such that σΘ=σΘ˜. □

Corollary 4.1.1 (see R. Olshen, [18: Theorem 3]).

If {X_n }_∈ℕis a P-exchangeable sequence of measurable maps from Ω into a complete, separable metric space, then there exists a real-valued random variable Θ on Ω such that {X_n}_∈ℕis P-conditionally i.i.d. given Θ.

Theorem 4.2.

Let {X_i}_i∈Ibe an infinite family of Σ-T-measurable maps from Ω into ϒ. Consider the following assertions:

1. {X_i}_i∈I is P-exchangeable.

2. There exists a σ-subalgebraFof Σ such that {X_i}_i∈IP-conditionally i.i.d. overF.

3. There exists a σ-subalgebraFof Σ and a family {Q_ω}_ω∈ΩofF-T-Markov kernels such that

   ∫FQωI(H)R(dω)=PF∩X−1(H)

   for every H ∈ T_I andF∈F, whereR:=P↾FandQωI, X are as in Lemma 4.1.

4. There exists a∑−Bd-measurable map Θ from Ω into ℝ^dsuch that {X_j}_i∈Iis P - conditionally i.i.d. given Θ.

Then (iii) ⇒ (i) ⇔ (ii) and (iv) ⇒ (i). If any one of conditions (i) to (iv) is satisfied, then all image measures P_Xi are equal.

Moreover, if P_Xi is perfect for any i ∈ I and T is countably generated, then assertions (i) to (iv) are all equivalent.

Proof. The equivalence (i) ⇔ (ii) follows by [7: Theorem 459B], while the implications (iii) ⇒ (i) and (iv) ⇒ (i) are evident.

Clearly, if assertion (i) or equivalently (ii) is satisfied then all P_Xi are equal and the same applies if (iii) or (iv) holds true.

Moreover, if every measure P_Xi is perfect and T is countably generated, then implications (ii) ⇒ (iii) and (i) ⇒ (iv) follow from Lemma 4.1 and Proposition 4.1, respectively. So, we get that assertions (i) to (iv) are all equivalent. □

Let {X_t}_t∈ℝ+be a family of Σ-T-measurable maps from Ω into ϒ. Suppose that Σ is countably generated, P is perfect, ϒ is a Polish space, {X_t}_t∈ℝ+is separable with separator ℚ₊and that its canonical filtration is right-continuous. Then each of the items (i) to (iv) of Theorem 4.2is equivalent to condition

1. there exist a d-dimensional random vector Θ and an rcp {P_θ}_θ∈ℝ^dof P over P_θ consistent with Θ such that {X_t}_t∈ℝ+is P_θ-i.i.d. for P_θ-a.a. θ ∈ ℝ^d.

Proof. It follows by our assumptions for P, Σ and ϒ, that given a d-dimensional random vector Θ, there exists an rcp {P_θ}_θ∈ℝ^d of P over P_Θ consistent with Θ. Thus, we may apply Proposition 3.2 to obtain that condition (v) is equivalent to (iv) of Theorem 4.2. The equivalence of all items (i) to (v) is immediate by Theorem 4.2. □

1. The assumption “P_Xi perfect” made in the last theorem is easily verified in the usual applications, since this is covered by the following facts: (α) If ϒ is a Polish space then each P_Xi is Radon (cf. e.g., [7: Proposition 434K(b)] and [7: Definition 411H(b)] for the definition of a Radon measure); hence perfect (cf. e.g., [7: Proposition 416W(a)]). (β) If P is perfect then each P_Xi is so (cf. e.g., [7: Proposition 451E(a)]). (γ) If Ω = ϒ^I, X_i(i ∈ I) are the canonical projections from Ω onto ϒ, and P is any probability measure on Σ := T_I then each P_Xi is perfect if and only if P is perfect (cf. e.g., [7: Theorem 454A(b)(iii)]).

2. To the best of our knowledge, the most general result concerning the equivalence of assertions (i) and (iii) of Theorem 4.2 is Theorem 1.1 from [12], which extends de Finetti’s Theorem, saying that for each infinite sequence {X_n}_n∈ℕ of random variables taking values in a standard Borel space ϒ (i.e. ϒ is isomorphic to some Borel-measurable subset of ℝ) assertions (i) and (iii) of Theorem 4.2 with {X_n}_n∈ℕ in the place of {X_i}_i∈I are equivalent. It is well-known that any Polish space is standard Borel; in particular, ℝ^d and ℝ^ℕ are such spaces.

3. There are measurable spaces (ϒ, T) satisfying the assumptions of Theorem 4.2, i.e., that T is countably generated and each P_Xi is perfect, which are not standard Borel spaces; hence Theorem 4.2 extends Theorem 1.1 from [12]. In fact, it is known that each uncountable analytic Hausdorff space (i.e., a non-empty topological Hausdorff space being a continuous image of the space ℕ^ℕ, cf. e.g., [7: Definition 423A]) has a non-Borel analytic subset (cf. e.g., [7: Proposition 423L]). It is also known that for each analytic Hausdorff space ϒ the Borel σ-algebra Bϒ is countably generated (cf. e.g., [7: 423X(d)]), and that any Borel probability measure on Bϒ is always inner regular with respect to compact sets (see [10: Chapter IV, Theorem 1, p. 195]); hence it is perfect (cf. e.g., [7: Proposition 451C]). Consequently, each uncountable analytic Hausdorff space has a subset satisfying the assumptions of Theorem 4.2, but not being a standard Borel space.

4. Restricting attention to measurable spaces (ϒ, T) satisfying the countability assumption of T as in Theorem 4.2, costs us some generality; for instance, the general compact Hausdorff space does not satisfy the countability assumption for T, and it is known that the equivalence of assertions (i) to (iii) of Theorem 4.2 is true for countable products of compact Hausdorff spaces (see [9] or [3]). More general, the equivalence of assertions (i) to (iii) of Theorem 4.2 is proven in [7: Theorem 459G] for uncountable products of general Hausdorff spaces. But all the above are specialized in the product situation of topological spaces, while assertions (i) to (iii) of Theorem 4.2 have the advantage of being free from any topological assumption as well as from any product situation.

The following definition of an MRP traces back to Huang [11: Section 1, Definition 3].

The counting process {N_t}_t∈ℝ+ is said to be a ν-mixed renewal process associated with Py˜y˜∈ϒ˜, if for every r ∈ ℝ and for all w₁,…, w_r ∈ ℝ condition

holds true, where Py˜y˜∈ϒ˜ is a family of probability measures on Σ and ν is a probability measure on B(ϒ˜):=σP•(E):E∈Σ such that for ν-a.a. y˜∈ϒ˜ the process {W_n}_n∈ℕ is Py˜-identically distributed.

In Huang’s definition it is assumed that {N_t}_t∈ℝ+ takes values only in ℕ₀, which is equivalent to the mild assumption that {N_t}_t∈ℝ+ has zero probability of explosion, that is P∪t∈(0,∞)Nt=∞=0 (cf. e.g., [21: Lemma 2.1.4]).

Note that in Huang’s [11] definition the assumption that {W_n}_n∈ℕ is Py∼-identically distributed for ν-a.a. y˜∈ϒ˜ is not written explicitly. But this assumption must be included there, since it is necessary for the validity of the Corollary in page 20 of [11], as it follows from Example 5 below.

In fact, consider the process {N_t}_t∈ℝ+ of Example 5, where the above assumption does not hold true, as well as Huang’s definition of a ν-MRP without the above assumption. Also note that q := P(Z < ∞) = 0 < 1, where Z is the almost sure limit of {N_t}_t∈ℝ+ as t → ∞. Assume, if possible, that the Corollary in [11: p. 20] holds true. Then conditional on the event {Z = ∞} the process {N_t}_t∈ℝ+ has the exchangeable property (E) (see [11: Definition 1] for the definition) implying that {W_n}_n∈ℕ is exchangeable, a contradiction to Example 5.

Consider the following assertions:

1. There exists a d-dimensional (d ∈ ℕ) random vector Θ such that {N_t}_t∈ℝ+is a (P, K(Θ))-MRP.

2. There exist a random vector Θ, an rcp {P_θ}_θ∈ℝ^dof P over P_Θ consistent with Θ, and a family {K(θ)}_θ∈ℝ^dofZ−B-Markov kernels for P_θ-a.a. θ ∈ ℝ^dsuch that the family {N_t}_t∈ℝ+is a (P_θ,K(θ))-RP.

3. The process {W_n}_n∈ℕis P-exchangeable.

4. There exist a σ-subalgebraFofΣand a family {Q_ω}_ω∈ΩofF−B-Markov kernels such that

   ∫FQωℕ(H)R(dω)=PF∩W−1(H)

   for everyH∈BℕandF∈F, whereR≔P↾F, W := (W₁, … W_n …) andQωℕdenotes the ℕ-fold product probability ⊗_n∈ℕP_nof copies P_n := Q_ω of Q_ω for n ∈ ℕ.

5. There exist a σ-subalgebraFof Σ, a subfield rcp {S_ω}_ω∈Ωfor P over the restrictionR≔P↾F, and a family {K(ω)}_ω∈ΩofF−B-Markov kernels such that for R-a.a. ω ∈ Ω the sequence {W_n}_n∈ℕis S_ω-independent and condition (S_ω)_Wn = K(ω) holds for each n ∈ ℕ.

6. There exist a setϒ˜, a familySy˜y˜∈ϒ˜; of probability measures on Σ and a probability measure ν onBϒ˜:=σS•(E):E∈Σsuch that {N_t}_t∈ℝ+is a ν-MRP associated withSy˜y˜∈ϒ.

Then the following implications hold true:

Proof. First note that the implications (i) ⟹ (iii) and (vi) ⟹ (iii) are obvious. The implication (ii) ⟹ (i) is immediate by Proposition 3.1, while the implication (iii) ⟹ (i) and the equivalence of (iii) and (iv) follow directly by Proposition 4.1 and Theorem 4.2, respectively, since for ϒ,T=ℝ,B every measure P_Wn on B is perfect and B is countably generated. The latter equivalence together with implication (vi) ⇒ (iii) yields (vi) ⇒ (iv).

Ad (ii) ⇒ (iv): If (ii) holds true, then there exists a P_Θ-null set H*∈Bd such that for any θ ∉ H_*, the process {W_n}_n∈ℕ is P_θ-exchangeable, implying together with property (d2) its P-exchangeability as well; hence (iii) or equivalently (iv) follows.

Ad (ii) ⇒ (v): Assume that (ii) is true. Putting S_ω(E) := P_Θ(E) for any ω ∈ Ω, E ∈ Σ and θ = Θ(ω), we clearly get that {S_ω}_ω∈Ω is a subfield rcp for P over R := P ↾ σ(Θ) such that the interarrival times W_n, n ∈ ℕ, are S_ω-i.i.d. with a common probability distribution K(ω) for any ω ∉ H_** := Θ⁻¹(H_*), where K(ω) := K(θ) for each ω ∈ Ω and Θ(ω)=θ∉H∗∈Bd. Since clearly H_** is an R-null set, it follows that {S_ω}_ω∈Ω, F:=σΘ and {K(ω)}_ω∈Ω satisfy assertion (v).

Ad (v) ⇒ (vi): Assume that (v) holds true and let F, {S_ω}_ω∈Ω and R be as in (v). Put ϒ˜:=Ω, Sy˜y˜∈ϒ˜:=Sωω∈Ω and B(ϒ˜):=σS•(E):E∈Σ. Then B(ϒ˜)⊆F and so, we may define the probability measure ν:=R↾B(ϒ˜). Since by (v) the process {W_n}_n∈ℕ is S_ω-i.i.d. for R-a.a. ω ∈ Ω, we get that it is Sy˜-i.i.d. for ν-a.a. y˜∈ϒ˜. The latter together with the definition of a subfield rcp yields that {N_t}_t∈ℝ+ is a ν-MRP associated with Sy˜y˜∈ϒ; hence assertion (vi) follows.

Moreover, if Σ is countably generated and P is perfect, the implication (iv) ⇒ (v) holds true. In fact, by Theorem 4.2, we obtain that assertion (iv) is equivalent with the fact that {W_n}_n∈ℕ is P-conditionally i.i.d. over F. But since Σ is countably generated and P is perfect, there exists a subfield rcp {S_ω}_ω∈Ω for P over R:=P↾F. Thus, we may apply Lemma 3.4(ii) Ψ,Z:=Ω,F and Θ := id_Ωto get (v).

Assuming now that (i) holds true, it follows again by our assumptions for Σ and P that there exists an rcp {P_θ}_θ∈ℝ^d of P over P_Θ consistent with Θ. So according to Proposition 3.1, we get that for P_Θ-a.a. θ ∈ ℝ^d the family {N_t}_t∈ℝ+ is a (P_θ, K(θ))-RP; hence (i) implies (ii).

Thus, assuming that Σ is countably generated and P is perfect, we obtain that items (i) to (vi) are all equivalent. This completes the whole proof. □

Note that the most important applications in Probability Theory are still rooted in the case of standard Borel spaces; hence of spaces satisfying always the assumptions of the above theorem concerning P and Σ.

Let ϒ := ℝ₊ and let Q˜ be a probability measure on Bϒ. Consider a subset B of ϒ such that Q˜*(B)=Q˜*(Bc)=1 where Q˜∗ is the outer measure induced by Q˜. Let T≔σBϒ,B and let Q: T → [0, 1] be the probability measure defined by

Then Q is non perfect. Set Ω := ϒ^ℕ, Σ := T_ℕ, P := Q^ℕ, W_n := π_n : Ω → ϒ, where π_n is the canonical projection for any n ∈ ℕ. Clearly, assertion (i) of Theorem 4.2 is satisfied by {W_n}_n∈ℕ; hence by Theorem 4.2, we equivalently get that (ii) holds true.

Assume that assertion (iii) of Theorem 4.2 is valid, i.e. that there exists a σ-subalgebra F of Σ and a family {Q_ω}_ω∈Ω of F-T-Markov kernels such that

for every H ∈ T_ℕ and A⊆F, where R:=P↾F and Qωℕ, W are as in Theorem 4.2. Then Qωℕω∈Ω is a subfield rcp for P over R; hence, we may and do assume that F is countably generated. Applying now a monotone class argument we deduce that there exists an R-null set N∈F such that for each A∈F condition Qωℕ(A)=1 holds true for any ω ∈ N^c ⋂ A.

But since Qωℕω∈Ω is a subfield rcp, we get for every F∈F that

implying that P(D) = 0, where D:=ω∈Ω:QωℕBℕ≠1.

Put E := D ⋃ N. For any ω ∈ E^c, we get Qωℕ({ω})=1 and QωℕBℕ=1 hence QωℕBℕ∩{ω}=1 implying Bℕ∩(ω}≠0 or ω ∈ B^ℕ. Thus, we get E^c ⊆ B^ℕ. But then Q˜∗Bc=1 yields P(E^c) = 0 or equivalently P(E) = 1, a contradiction.

Assume now that assertion (v) of Corollary 4.2.1 is valid, i.e., that there exist a d-dimensional random vector Θ and an rcp {P_θ}_θ∈ℝ^d of P over P_Θ consistent with Θ such that {W_n}_n∈ℕ is P_θ-i.i.d. for P_Θ-a.a. θ ∈ ℝ^d. For any fixed A ∈Σ define R˜•(A):Ω→[0,1] by R˜ω(A):=P•(A)∘Θ(ω). Then R˜ωω∈Ω is a subfield rcp for P over P ↾ σ(Θ), a contradiction according to the above arguments.

It follows an example to show that the countability assumption for T or Σ in Corollary 4.2.1 is essential for the validity of the equivalences of (i) to (v) in Theorem 4.2 and Corollary 4.2.1.

Let ϒ := ℝ₊, let Q˜ be a probability measure on Bϒ, let T be the completion of Bϒ with respect to Q˜ and let Q be the completion of Q˜. Put Ω := ϒ^ℕ, Σ˜:=Tℕ and P˜:=Qℕ. Denote by Σ the completion of Σ˜ with respect to P˜ and by P the completion of P˜. Then P is a perfect probability measure on Σ (cf. e.g., [7: Proposition 451G and Theorem 451J]) but Σ and T are not countably generated.

Put W_n := π_n : Ω → ϒ for any n ∈ ℕ. Clearly, {W_n}_n∈ℕ is P-exchangable, that is, assertion (i) of Theorem 4.2 is valid; hence assertion (ii) of the same theorem is also valid.

Assume now that assertion (v) of Corollary 4.2.1 is valid. It then follows that there exists a d-dimensional random vector Θ and an rcp {P_θ}_θ∈ℝ^d of P over P_Θ consistent with Θ, implying the existence of a subfiled rcp {R_ω}_ω∈Ω of P over P ↾ σ(Θ), where each function R_•(A): Ω → [0, 1] is defined by R_ω(A) := (P_•(A) ∘ Θ)(ω) for any fixed A ∈ Σ. Since σ(Θ) is countably generated, it follows as in Example 1 that there exists a set N ∈ σ(Θ) such that P(N) = 0 and for any A ∈ σ(Θ) condition R_ω(A) = 1 holds true for any ω ∈ N^c ⋂ A.

Choose a set D ⊆ N^c such that D ∉ σ(Θ) but D ∈ Σ. Then for each ω ∉ N, we obtain

and

Thus, D = N^c ⋂ {ω ∈ Ω : R_ω(D) = 1} ∈ σ(Θ), which is impossible by the choice of D; hence assertion (v) of Corollary 4.2.1 is not valid.

The next counterexample shows that there exists a non perfect probability space (Ω, Σ, P) and a counting process {N_t}_t∈ℝ⁺ on it satisfying conditions (i), (iii), (iv) and (vi) of Theorem 4.3 but not (ii) and (v); hence the perfectness of P is an essential assumption.

Let ϒ := ℝ₊, let Q be a probability measure on T:=Bϒ, and let (Ω,Σ˜,P˜):=Υℕ,Tℕ,Qℕ. Consider a subset B of ϒ^ℕ such that P˜∗(B)=P˜∗Bc=1 the σ-algebra Σ:=σ({Σ˜∪B}) and the non perfect probability measure P on Σ defined by means of

Let W_n := π_n : Ω → ϒ be the canonical projection for any n ∈ ℕ. Clearly, assertion (iii) of Theorem 4.3 is satisfied by {W_n}_n∈ℕ, and so, we equivalently get that assertions (iv) and (i) of the same theorem also hold true. Furthermore, it can be easily seen that (iv) ⇒ (vi) as in Theorem 4.3; hence (iv) ⇔ (vi).

Applying similar arguments with those in Example 1, we obtain that assertions (v) and (ii) of Theorem 4.3 are not valid.

The next counterexample shows that the countability assumption for Σ in Theorem 4.3 is essential for the equivalence of items (i) to (vi) of Theorem 4.3.

Let (ϒ, T, Q) and (Ω, Σ˜, P˜) be as in Example 1, let Σ be the completion of Σ˜ with respect to P˜ and P the completion of P˜, and let {W_n}_n∈ℝ be as in Example 1. Then P is a perfect probability measure on Σ but Σ is not countably generated.

It then follows that assertion (iii) of Theorem 4.3 holds true, while by the same theorem, we get that its assertions (i), (iii) and (iv) are all equivalent. Furthermore, it can be easily shown that (iv) implies (vi). But using similar arguments as in Example 2, we get that assertion (ii) of Theorem 4.3 is not valid.

Assume that assertion (v) of Theorem 4.3 holds true. It then follows that there exist a σ-subalgebra ℱ of Σ and a subfield rcp {S_ω}_ω∈Ω of P over R:=P↾ℱ.

There exists a countably generated σ-subalgebra A of F such that {S_ω}_ω∈Ω satisfies

and

In fact, applying similar arguments with those in the proof of Proposition 4.1, we get a countably generated σ-subalgebra A˜ of F such that for any fixed H˜∈Σ˜ the function S•(H˜) is A˜-measurable. Since for R-a.a. ω∈Ω the measures P and S_ω have the same null sets, we obtain that for any H ∈Σ there exist sets H˜∈Σ˜, M_H ∈ Σ and NH∈F∩Σ0 such that H=H˜∪MH and S_ω(M_H) =P(M_H) = 0 for all ω ∉ N_H; hence {S_ω}_ω∈Ω satisfies (5.2) with A˜ in the place of A. Applying a monotone class argument, we get a P-null set N1∈F such that for any fixed H˜∈A˜ the function S•(H˜)↾N1c is A˜-measurable.

Put A:=σA˜∪N1⊆F. It then follows that A is countably generated and {S_ω}_ω∈Ω satisfies (5.1) and (5.2). In particular, for any fixed H∈A the function S_•(H) ↾ (N₁)^c is A-measurable.

Applying now a monotone class argument, we obtain a P-null set N2∈A such that for any H∈A and ω ∈ N^c ⋂H, where N := N₁ ⋃ N₂, condition S_ω(H) = 1 holds true.

Choose a set D ⊆ N^c such that D∉A but D ∉ Σ. Following the same reasoning as in Example 2, we deduce thatD∈A which is impossible by the choice of D; hence assertion (v) of Theorem 4.3 is not valid.

Throughout what follows, we putΩ˜:=ℝℕ, Ω:=Ω˜×ℝd, Σ˜:=B(Ω˜)andΣ:=Σ˜⊗Zfor simplicity. The next result extends Theorem 3.1 from [14], which provides a construction for MPPs.

Following the reasoning of Theorem 3.1 from [14] but with ℝ^d, B and Q_n(θ) = K(θ) in the place of ϒ := (0, ∞), Bϒ and Q_n(θ) = Exp(θ), respectively, the existence of (P, K(Θ))- MRPs with prescribed distributions for their interarrival processes as well as for the parameter Θ is proven.

In fact, fix on arbitrary θ ∈ ℝ^d and put P˜θ:=⊗n∈ℕQn(θ). Since by assumption, for any fixed B∈B each function Q_n(·)(B) is Z-measurable, it follows by a monotone class argument that the same holds true for the function P˜•(E) for fixed E∈Σ˜.

For each θ ∈ ℝ^d put Pθ:=P˜θ⊗δθ where δ_θ is the Dirac probability measure on Z, and for each n ∈ ℕ set W_n := π_n, where π_n is the canonical projection from Ω onto ℝ. Put now

where E^θ := {ω ∈ Ω : (ω, θ) ∈ E}. Then P is a probability measure on Σ such that {P_θ}_θ∈ℝ^d is an rcp of P over μ consistent with π_ℝ^d, where π_ℝ^d is the canonical projection from Ω onto ℝ^d (see [14: proof of Theorem 3.1]). Furthermore, it can be proven that for all θ ∈ ℝ^d the process {W_n}_n∈ℕ is p_θ-independent and (p_θ)_Wn = K(θ) for each n ∈ ℕ. Clearly, putting Θ := π_ℝ^d, we get P_θ = μ.

It then follows that the counting process {N_t}_t∈ℝ induced by {W_n}_n∈ℕ (cf. e.g., [21: Theorem 2.1.1]) is a (P_θ, K(θ))-RP for all θ ∈ ℝ^d; hence by Proposition 3.1 it is a (P, K(Θ))-MRP.