Appendix A
Proof of Theorem 2.1
We condition both sums, \( Z_{1} (t) \) and \( Z_{2} (t) \), on \( N(t) \) and use Lemma 2.1 by following the same steps as described in Léveillé and Hamel [15].Hence, if we let \( W_{k,i} = X_{k} 1_{{\{ 1\} }} (i) + Y_{k} 1_{{\{ 2\} }} (i) \) and \( W_{i} = X1_{{\{ 1\} }} (i) + Y1_{{\{ 2\} }} (i) \), we obtain
$$ \begin{aligned} & E\left[ {E\left[ {\left. {Z_{i} (t)} \right|N(t)} \right]} \right] \\ & \quad = \sum\limits_{n = 1}^{\infty } {\sum\limits_{k = 1}^{\infty } {E\left[ {\left. {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{i} (T_{k} + \xi_{k} + \zeta_{k} )W_{k,i} } \right|N(t) = n} \right]P(N(t) = n)} } \\ & \quad = \sum\limits_{n = 1}^{\infty } {\sum\limits_{k = 1}^{\infty } {\int\limits_{0}^{t} {E\left[ {\left. {I(u,\xi_{k} ,\zeta_{k} ,t)D_{i} \left( {u + \xi_{k} + \zeta_{k} } \right)W_{k,i} } \right|N(t) = n} \right]dF_{{\left. {T_{k} } \right|N(t) = n}} (u)} P(N(t) = n)} } \\ & \quad = \sum\limits_{k = 1}^{\infty } {\sum\limits_{n = k}^{\infty } {\int\limits_{0}^{t} {E[I(u,\xi_{k} ,\zeta_{k} ,t)D_{i} (u + \xi_{k} + \zeta_{k} )W_{k,i} ]P(\tilde{N}(\varLambda (t) - \varLambda (u)) = n - k)dF^{*k} } (\varLambda (u))} } \\ & \quad = \sum\limits_{k = 1}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {I(u,v,w,t)E[D_{i} (u + v + w)]E\left[ {\left. {W_{k,i} } \right|\zeta = w} \right]dF_{\xi } (v)dF_{\zeta } (w)dF^{*k} (\varLambda (u))} } } } \\ & \quad = \int\limits_{0}^{\infty } {E\left[ {\left. {W_{i} } \right|\zeta = w} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {I(u,v,w,t)E[D_{i} (u + v + w)]d\tilde{m}(\varLambda (u))dF_{\xi } (v)dF_{\zeta } (w)} } } . \\ \end{aligned} $$
The result follows by adding both conditional expectations. \( \square \)
Proof of Theorem 2.2
First we multiply the sums between them, then we separate each term in a suitable way, i.e. for indices \( k = j \), \( \;k < j \) and \( k > j \). Hence, we get the following expression
$$ \begin{aligned} Z^{2} (t) & = \sum\limits_{k = 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{1}^{2} (T_{k} + \xi_{k} + \zeta_{k} )X_{k}^{2} } \\ & \quad + \sum\limits_{k = 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{2}^{2} (T_{k} + \xi_{k} + \zeta_{k} )Y_{k}^{2} } \\ & \quad + 2\left\{ {\sum\limits_{k = 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{2} (T_{k} + \xi_{k} + \zeta_{k} )X_{k} Y_{k} } } \right. \\ & \quad + \sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} X_{j} } } \\ & \quad + \sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{2} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{2} (T_{j} + \xi_{j} + \zeta_{j} )Y_{k} Y_{j} } } \\ & \quad + \sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{2} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} Y_{j} } } \\ & \quad \left. { + \sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{2} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{j} Y_{k} } } } \right\}. \\ \end{aligned} $$
Again using Lemma 2.1, we obtain for the expectation of the first two terms,
$$ \begin{aligned} & E\left[ {\sum\limits_{k = 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{i}^{2} (T_{k} + \xi_{k} + \zeta_{k} )W_{k,i}^{2} } } \right] \\ & = \sum\limits_{n = 1}^{\infty } {\sum\limits_{k = 1}^{n} {E\left[ {\left. {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{i}^{2} (T_{k} + \xi_{k} + \zeta_{k} )W_{k,i}^{2} } \right|N(t) = n} \right]P(N(t) = n)} } \\ & = \sum\limits_{n = 1}^{\infty } {\sum\limits_{k = 1}^{n} {\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {I(u,v,w,t)E[D_{i}^{2} (u + v + w)]E\left[ {\left. {W_{i}^{2} } \right|\zeta = w} \right]dF_{{T_{k} \left| {N(t) = n} \right.}} (u)dF_{\xi } (v)dF_{\zeta } (w) \times P(N(t) = n)} } } } } \\ & = \sum\limits_{k = 1}^{\infty } {\sum\limits_{n = k}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {I\left( {u,v,w,t} \right)E[D_{i}^{2} (u + v + w)]E\left[ {\left. {W_{i}^{2} } \right|\zeta = w} \right]P(\tilde{N}(\varLambda (t) - \varLambda (u)) = n - k) \times dF^{*k} (\varLambda (u))dF_{\xi } (v)dF_{\zeta } (w)} } } } } \\ & = \int\limits_{0}^{\infty } {E\left[ {\left. {W_{i}^{2} } \right|\zeta = w} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {I(u,v,w,t)E[D_{i}^{2} (u + v + w)]d\tilde{m}(\varLambda (u))dF_{\xi } (v)dF_{\zeta } (w)} } } . \\ \end{aligned} $$
The expectation of the third term yields,
$$ \begin{aligned} & E\left[ {\sum\limits_{k = 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{2} (T_{k} + \xi_{k} + \zeta_{k} )X_{k} Y_{k} } } \right] \\ & \quad = \int\limits_{0}^{\infty } {E\left[ {\left. {XY} \right|\zeta = w} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {I(u,v,w,t)E[D_{1} (u + v + w)D_{2} (u + v + w)]d\tilde{m}(\varLambda (u))dF_{\xi } (v)dF_{\zeta } (w)} } } . \\ \end{aligned} $$
Again using Lemma 2.1, we obtain for the expectation of the fourth and fifth terms,
$$ \begin{aligned} & E\left[ {\sum\limits_{k = 1}^{N\left( t \right) - 1} {\sum\limits_{j = k + 1}^{N\left( t \right)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{i} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{i} (T_{j} + \xi_{j} + \zeta_{j} )W_{k,i} W_{j,i} } } } \right] \\ & \quad = \sum\limits_{n = 2}^{\infty } {\sum\limits_{k = 1}^{n - 1} {\sum\limits_{j = k + 1}^{n} {\int\limits_{0}^{t} {\int\limits_{u}^{t} {E[I(u,\xi_{k} ,\zeta_{k} ,t)D_{i} (u + \xi_{k} + \zeta_{k} )I(z,\xi_{j} ,\zeta_{j} ,t)D_{i} (z + \xi_{j} + \zeta_{j} )W_{k,i} W_{j,i} ]} } } } } \\ & \quad \times dF_{{T_{k} ,T_{j} |N(t) = n}} (u,z)P(N(t) = n) \\ & \quad = \sum\limits_{k = 1}^{\infty } {\sum\limits_{j = k + 1}^{\infty } {\sum\limits_{n = j}^{\infty } {\int\limits_{0}^{t} {\int\limits_{u}^{t} {E[I(u,\xi_{k} ,\zeta_{k} ,t)D_{i} (u + \xi_{k} + \zeta_{k} )I(z,\xi_{j} ,\zeta_{j} ,t)D_{i} (z + \xi_{j} + \zeta_{j} )W_{k,i} W_{j,i} ]} } } } } \\ & \quad \times P(\tilde{N}(\varLambda (t) - \varLambda (z)) = n - j)dF^{*(j - k)} (\varLambda (z) - \varLambda (u))dF^{*k} (\varLambda (u)) \\ & \quad = \sum\limits_{k = 1}^{\infty } {\sum\limits_{j = k + 1}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{t - u} {I(u,v,w,t)I(u + u^{\prime } ,v^{\prime } ,w^{\prime } ,t)E[D_{i} (u + v + w)D_{i} (u + u^{\prime } + v^{\prime } + w^{\prime } )]} } } } } } } } \\ & \quad \quad \times E\left[ {\left. {W_{i} } \right|\zeta = w} \right]E\left[ {\left. {W_{i} } \right|\zeta = w^{\prime}} \right]dF^{*(j - k)} (\varLambda (u + u^{\prime } ) - \varLambda (u))dF^{*k} (\varLambda (u)) \\ & \quad \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime } )dF_{\zeta } (w)dF_{\zeta } (w^{\prime } ) \\ \end{aligned} $$
which yields,
$$ \begin{aligned} & = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. {W_{i} } \right|\zeta = w} \right]E\left[ {\left. {W_{i} } \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{t - u} {I(u,v,w,t)I(u + u^{\prime } ,v^{\prime } ,w^{\prime } ,t)} } } } } } \\ & \quad \times E[D_{i} (u + v + w)D_{i} (u + u^{\prime } + v^{\prime } + w^{\prime } )]d\tilde{m}(\varLambda (u + u^{\prime } ) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ & \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime } )dF_{\zeta } (w)dF_{\zeta } (w^{\prime } ). \\ \end{aligned} $$
The expectation of the sixth term yields,
$$ \begin{aligned} & E\left[ {\sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{2} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} Y_{j} } } } \right] \\ & \quad = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. Y \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{t - u} {I(u,v,w,t)I(u + u^{\prime},v^{\prime},w^{\prime},t)} } } } } } \\ & \quad \quad \times E[D_{1} (u + v + w)D_{2} (u + u^{\prime} + v^{\prime} + w^{\prime})]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ \quad \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime})dF_{\zeta } (w)dF_{\zeta } (w^{\prime}). \\ \end{aligned} $$
And finally, the expectation of the seventh term yields,
$$ \begin{aligned} & E\left[ {\sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)D_{2} (T_{k} + \xi_{k} + \zeta_{k} )I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)D_{1} (T_{j} + \xi_{j} + \zeta_{j} )Y_{k} X_{j} } } } \right] \\ & \quad = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. Y \right|\zeta = w} \right]E\left[ {\left. X \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{t - u} {I(u,v,w,t)I(u + u^{\prime},v^{\prime},w^{\prime},t)} } } } } } \\ & \quad \times E\left[ {D_{2} (u + v + w)D_{1} (u + u^{\prime} + v^{\prime} + w^{\prime})} \right]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ & \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime})dF_{\zeta } (w)dF_{\zeta } (w^{\prime}). \\ \end{aligned} $$
For the remainder of the proof, we have simply to gather the suitable terms. □
Proof of Theorem 2.3
We first isolate \( Z^{2} (t) \) from the product \( Z(t)Z(t + h) \), then we multiply the remaining sums between them, we separate each term in a suitable way, we condition each sum on \( N(t) \) and \( N(t + h) \) where it is appropriate. Hence, we first get
$$ \begin{aligned} & Z(t)Z(t + h) = Z^{2} (t) + [Z_{1} (t) + Z_{2} (t)]\left\{ {\sum\limits_{k = 1}^{N(t)} {[I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)]D_{1} (T_{k} + \xi_{k} + \zeta_{k} )X_{k} } } \right. \\ & \quad + \sum\limits_{k = N(t) + 1}^{N(t + h)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )X_{k} } \\ & \quad + \sum\limits_{k = 1}^{N(t)} {[I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)]D_{2} (T_{k} + \xi_{k} + \zeta_{k} )Y_{k} } \\ & \quad \left. { + \sum\limits_{k = N(t) + 1}^{N(t + h)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h)D_{2} (T_{k} + \xi_{k} + \zeta_{k} )Y_{k} } } \right\}. \\ \end{aligned} $$
For the expectation of the first term (with factor \( Z_{1} (t) \)), we get
$$ \begin{aligned} & E\left[ {Z_{1} (t)\sum\limits_{k = 1}^{N(t)} {[I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)]D_{1} (T_{k} + \xi_{k} + \zeta_{k} )X_{k} } } \right] \\ & \quad = E\left[ {\sum\limits_{k = 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)[I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)]D_{1}^{2} [T_{k} + \xi_{k} + \zeta_{k} ]X_{k}^{2} } } \right. \\ & \quad \quad + \sum\limits_{k = 1}^{N(t) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)[I(T_{j} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)]D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} X_{j} } } \\ & \quad \quad + \sum\limits_{j = 1}^{N(t) - 1} {\left. {\sum\limits_{k = j + 1}^{N\left( t \right)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)[I(T_{j} ,\xi_{j} ,\zeta_{j} ,t + h) - I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)]D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} X_{j} } } \right]} \\ & \quad = E\left[ {\sum\limits_{k = 1}^{N\left( t \right) - 1} {\sum\limits_{j = k + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)\left[ {I(T_{j} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)} \right]D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} X_{j} } } } \right. \\ & \quad \quad + \sum\limits_{j = 1}^{N(t) - 1} {\left. {\sum\limits_{k = j + 1}^{N(t)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)[I(T_{j} ,\xi_{j} ,\zeta_{j} ,t + h) - I(T_{j} ,\xi_{j} ,\zeta_{j} ,t)]D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{k} X_{j} } } \right]} \\ & = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. X \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{t - u} {I(u,v,w,t)[I(u + u^{\prime } ,v^{\prime } ,w^{\prime } ,t + h) - I(u + u^{\prime } ,v^{\prime } ,w^{\prime } ,t)]} } } } } } \\ & \quad \quad \times E[D_{1} (u + v + w)D_{1} (u + u^{\prime} + v^{\prime} + w^{\prime})]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ & \quad \quad \times dF_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tau } }} (v)dF_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tau } }} (v^{\prime})dF_{{\tilde{\tau }}} (w)dF_{{\tilde{\tau }}} (w^{\prime}) \\ & \quad \quad + \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. X \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{{t - u^{\prime}}} {I(u + u^{\prime},v,w,t)[I(u^{\prime},v^{\prime},w^{\prime},t + h) - I(u^{\prime},v^{\prime},w^{\prime},t)]} } } } } } \\ \quad \quad \times E[D_{1} (u + u^{\prime} + v + w)D_{1} (u^{\prime} + v^{\prime} + w^{\prime})]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u^{\prime}))d\tilde{m}(\varLambda (u^{\prime})) \\ \quad \quad \times dF_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tau } }} (v)dF_{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\tau } }} (v^{\prime})dF_{{\tilde{\tau }}} (w)dF_{{\tilde{\tau }}} (w^{\prime}). \\ \end{aligned} $$
The expectation of the second term (with factor \( Z_{1} (t) \)) yields,
$$ \begin{aligned} & E\left[ {Z_{1} (t)\sum\limits_{j = N(t) + 1}^{N(t + h)} {I(T_{j} ,\xi_{j} ,\zeta_{j} ,t + h)D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{j} } } \right] \\ & \quad = E\left[ {\sum\limits_{k = 1}^{N(t)} {\sum\limits_{j = N(t) + 1}^{N(t + h)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)I(T_{j} ,\xi_{j} ,\zeta_{j} ,t + h)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )D_{1} (T_{j} + \xi_{j} + \zeta_{j} )X_{j} } } } \right] \\ & \quad = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. X \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{t - u}^{t + h - u} {I(u,v,w,t)I(u + u^{\prime},v^{\prime},w^{\prime},t + h)} } } } } } \\ & \quad \quad \times E[D_{1} (u + v + w)D_{1} (u + u^{\prime} + v^{\prime} + w^{\prime})]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ & \quad \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime})dF_{\zeta } \left( w \right)dF_{\zeta } (w^{\prime}). \\ \end{aligned} $$
Similarly to the two preceding expectations (with factor \( Z_{1} (t) \)), the third term yields,
$$ \begin{aligned} & E\left[ {Z_{1} (t)\sum\limits_{k = 1}^{N(t)} {\left[ {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h) - I(T_{k} ,\xi_{k} ,\zeta_{k} ,t)} \right]D_{2} (T_{k} + \xi_{k} + \zeta_{k} )Y_{k} } } \right] \\ & \quad = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. Y \right|\tilde{\tau } = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{t - u} {I(u,v,w,t)[I(u + u^{\prime},v^{\prime},w^{\prime},t + h) - I(u + u^{\prime},v^{\prime},w^{\prime},t)]} } } } } } \\ & \quad \quad \times E\left[ {D_{1} (u + v + w)D_{2} (u + u^{\prime} + v^{\prime} + w^{\prime})} \right]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ & \quad \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime})dF_{\zeta } (w)dF_{\zeta } (w^{\prime}) \\ & \quad + \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. X \right|\tilde{\tau } = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{0}^{{t - u^{\prime}}} {I(u + u^{\prime},v,w,t)[I(u^{\prime},v^{\prime},w^{\prime},t + h) - I(u^{\prime},v^{\prime},w^{\prime},t)]} } } } } } \\ & \quad \quad \times E[D_{1} (u + u^{\prime} + v + w)D_{2} (u^{\prime} + v^{\prime} + w^{\prime})]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u^{\prime}))d\tilde{m}(\varLambda (u^{\prime})) \\ & \quad \quad {\kern 1pt} \times dF_{\xi } (v)dF_{\xi } (v^{\prime})dF_{\zeta } (w)dF_{\zeta } (w^{\prime}), \\ \end{aligned} $$
and the fourth term yields,
$$ \begin{aligned} & E\left[ {Z_{1} (t)\sum\limits_{k = N(t) + 1}^{N(t + h)} {I(T_{k} ,\xi_{k} ,\zeta_{k} ,t + h)D_{1} (T_{k} + \xi_{k} + \zeta_{k} )X_{k} } } \right] \\ & \quad = \int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {E\left[ {\left. X \right|\zeta = w} \right]E\left[ {\left. Y \right|\zeta = w^{\prime}} \right]\int\limits_{0}^{\infty } {\int\limits_{0}^{\infty } {\int\limits_{0}^{t} {\int\limits_{t - u}^{t + h - u} {I(u,v,w,t)I(u + u^{\prime},v^{\prime},w^{\prime},t + h)} } } } } } \\ & \quad \quad \times E\left[ {D_{1} (u + v + w)D_{2} (u + u^{\prime} + v^{\prime} + w^{\prime})} \right]d\tilde{m}(\varLambda (u + u^{\prime}) - \varLambda (u))d\tilde{m}(\varLambda (u)) \\ & \quad \quad \times dF_{\xi } (v)dF_{\xi } (v^{\prime})dF_{\zeta } (w)dF_{\zeta } (w^{\prime}). \\ \end{aligned} $$
The same calculations can be done similarly with the factor \( Z_{2} (t) \) and, as in Theorem 2.2, the remainder of the proof simply consists to gather the suitable terms. \( \square \)
