Erratum - Some Characterizations of Mixed Poisson Processes

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

“…Since tP θ u θPD is a rcp of P over µ consistent with Θ by (b), applying [12], Lemma 4.1, along with [13], we obtain that W and X are P -conditionally independent, i.e. P satisfies condition (a1).…”

A characterization of martingale-equivalent compound mixed Poisson process

“…Since S has Q θ -stationary independent increments for any θ / ∈ C ( * ) , it follows by [12], Corollary 4.2 together with [14] that it has Q-conditionally stationary independent increments, implying for all s, t ∈ R + with s ≤ t that S t − S s is Q-conditionally independent of F S s ; hence of F s (see [12], Lemma 4.7). The latter together with [3], Section 7.3, Theorem 1 implies that condition…”

“…[18], Theorem 2.1.1 and Lemma 2.1.2). The equivalence (a) ⇐⇒ (c) follows by [12], Lemma 4.1 together with [14] Ad (ii): Assume that (a2) holds true. Then X is P -independent if and only if it is Pconditionally independent.…”

“…it follows that condition(14) holds true if and only if θ > 1/2. Furthermore,E Q [N 1 ] = E Q [Θ 2 ] = E P [Θ 2 ξ(Θ)] = (81/24)Γ(5) = 81 and E Q [X 1 ] = E P [X 2 1 ]/E P [X 1 ] = 10, implying p…”

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

“…Since {P θ } θ∈D is a rcp of P over µ consistent with Θ by (b), applying Lyberopoulos and Macheras (2012) Lemma 4.1 along with Lyberopoulos and Macheras (2014), we obtain that W and X are P -conditionally independent, i.e., P satisfies condition (a1). Furthermore, condition (A.3.1) again together with the fact that {P θ } θ∈D is a rcp of P over µ consistent with Θ by (b), implies that P Xn = R and condition (a2) is satisfied by P .…”