Appendix 1
In this Appendix, proof of Proposition 1 is presented.
Proof
We use method of mathematical induction to prove (7) and (8) for \(t=0,1,\ldots ,T-1\).
In the last trading period T, optimal strategy is to execute all the remaining shares to complete the trading. Thus
$$\begin{aligned} S_T= & {} W_T \nonumber \\= & {} \delta _{P,0}P_{T-1}+\delta _{PP,0}P_{T-2}+\delta _{W,0}W_{T}+\delta _{X,0}X_{T}+\delta _{XX,0}X_{T-1} \end{aligned}$$
(55)
where \(\delta _{P,0}=0,~\delta _{PP,0}=0,~\delta _{W,0}=1,~\delta _{X,0}=0,~\delta _{XX,0}=0\).
By (5), we get
$$\begin{aligned}&V_T(P_{T-1},P_{T-2},X_T,X_{T-1},W_T) \nonumber \\&\quad =u_1P_{T-1}W_T+u_2P_{T-2}W_T+\theta W_T^2+\gamma X_TW_T \nonumber \\&\quad =a_0P_{T-1}^2+b_0P_{T-2}^2+c_0W_{T}^2+d_0X_{T}^2+e_0X_{T-1}^2+f_0P_{T-1}P_{T-2}+g_0P_{T-1}W_{T}\nonumber \\&\qquad +h_0P_{T-1}X_{T} +i_0P_{T-1}X_{T-1}+j_0P_{T-2}W_{T}+k_0P_{T-2}X_{T}+l_0P_{T-2}X_{T-1}\nonumber \\&\qquad +m_0W_{T}X_{T}+n_0W_{T}X_{T-1} +o_0X_{T}X_{T-1}+p_0, \end{aligned}$$
(56)
where \(a_0=0,~b_0=0,~c_0=\theta ,~d_0=0,~e_0=0,~f_0=0,~g_0=u_1,~h_0=0,~i_0=0,~j_0=u_2,~k_0=0,~l_0=0\), \(m_0=\gamma ,~n_0=0,~o_0=0,~p_0=0\).
Let following equation holds true.
$$\begin{aligned}&V_{T-t+1}(P_{T-t},P_{T-t-1},X_{T-t+1},X_{T-t},W_{T-t+1})\nonumber \\&\quad =a_{t-1}P_{T-t}^2+b_{t-1}P_{T-t-1}^2+c_{t-1}W_{T-t+1}^2+d_{t-1}X_{T-t+1}^2+e_{t-1}X_{T-t}^2\nonumber \\&\qquad +f_{t-1}P_{T-t}P_{T-t-1} +g_{t-1}P_{T-t}W_{T-t+1}+h_{t-1}P_{T-t}X_{T-t+1}\nonumber \\&\qquad +i_{t-1}P_{T-t}X_{T-t}+j_{t-1}P_{T-t-1}W_{T-t+1} +k_{t-1}P_{T-t-1}X_{T-t+1}+l_{t-1}P_{T-t-1}X_{T-t}\nonumber \\&\qquad +m_{t-1}W_{T-t+1}X_{T-t+1}+n_{t-1}W_{T-t+1}X_{T-t} +o_{t-1}X_{T-t+1}X_{T-t}+p_{t-1} \end{aligned}$$
(57)
By Eq. (6), we have
$$\begin{aligned}&V_{T-t}(P_{T-t-1},P_{T-t-2},X_{T-t},X_{T-t-1},W_{T-t}) \nonumber \\&\quad =\min _{S_{T-t}}~E_{T-t}[P_{T-t}S_{T-t}+V_{T-t+1}(P_{T-t},P_{T-t-1},X_{T-t+1},X_{T-t},W_{T-t+1})] \nonumber \\&\qquad (\hbox {Putting value of }V_{T-t+1}\hbox { from Eq.} \hbox { (57)}) \nonumber \\&\quad =\min _{S_{T-t}}~E_{T-t}[P_{T-t}S_{T-t}+a_{t-1}P_{T-t}^2+b_{t-1}P_{T-t-1}^2+c_{t-1}W_{T-t+1}^2\nonumber \\&\qquad +d_{t-1}X_{T-t+1}^2 +e_{t-1}X_{T-t}^2+f_{t-1}P_{T-t}P_{T-t-1}+g_{t-1}P_{T-t}W_{T-t+1} \nonumber \\&\qquad +h_{t-1}P_{T-t}X_{T-t+1}+i_{t-1}P_{T-t}X_{T-t} +j_{t-1}P_{T-t-1}W_{T-t+1} \nonumber \\&\qquad +k_{t-1}P_{T-t-1}X_{T-t+1}+l_{t-1}P_{T-t-1}X_{T-t}+m_{t-1}W_{T-t+1}X_{T-t+1}\nonumber \\&\qquad +n_{t-1}W_{T-t+1}X_{T-t}+o_{t-1}X_{T-t+1}X_{T-t}+p_{t-1} ] \nonumber \\&\qquad (\hbox {Using Eqs. (3) and (4), first we put values of }P_{T-t} \hbox { and }X_{T-t+1}\hbox {, respectively.} \nonumber \\&\qquad \hbox {Further we use relation }W_{T-t+1}=W_{T-t}-S_{T-t},\hbox { and rearrange terms as a} \nonumber \\&\qquad \hbox { quadratic function in }S_{T-k}.) \nonumber \\&\quad =\min _{S_{T-t}}~E_{T-t}\Big [\Big \{\theta +a_{t-1}\theta ^2+c_{t-1}-g_{t-1}\theta \Big \}S_{T-t}^2 +\Big \{(2a_{t-1}u_1\theta +f_{t-1}\theta \nonumber \\&\qquad +u_1-j_{t-1}-g_{t-1}u_1)P_{T-t-1}+(2a_{t-1}u_2\theta +u_2-g_{t-1}u_2) P_{T-t-2}\gamma \nonumber \\&\qquad +(g_{t-1}\theta -2c_{t-1})W_{T-t}+(2a_{t-1}\theta +\gamma -g_{t-1}\gamma +i_{t-1}\theta -m_{t-1}\rho _1-n_{t-1})X_{T-t}\nonumber \\&\qquad +(h_{t-1}\theta \rho _2-m_{t-1}\rho _2)X_{T-t-1}\Big \}S_{T-t}+(a_{t-1}u_1^2+b_{t-1}+f_{t-1}u_1)P_{T-t-1}^2\nonumber \\&\qquad +(a_{t-1}u_2^2)P_{T-t-2}^2+(c_{t-1})W_{T-t}^2+(a_{t-1}\gamma ^2+d_{t-1}\rho _1^2+e_{t-1}\nonumber \\&\qquad +h_{t-1}\gamma \rho _1+i_{t-1}\gamma +o_{t-1}\rho _1)X_{T-t}^2 +(d_{t-1}\rho _2^2)X_{T-t-1}^2+(2a_{t-1}u_1u_2\nonumber \\&\qquad +f_{t-1}u_2)P_{T-t-1}P_{T-t-2} +(g_{t-1}u_1+j_{t-1})P_{T-t-1}W_{T-t}+(2a_{t-1}u_1\gamma +f_{t-1}\gamma \nonumber \\ {}&\qquad +h_{t-1}u_1\rho _1+i_{t-1}u_1+k_{t-1}\rho _1+l_{t-1})P_{T-t-1}X_{T-t}\nonumber \\&\qquad +(h_{t-1}u_1\rho _2+k_{t-1}\rho _2)P_{T-t-1}X_{T-t-1}\nonumber \\ {}&\qquad +(g_{t-1}u_2)P_{T-t-2}W_{T-t} +(2a_{t-1}u_2\gamma +h_{t-1}u_2\rho _1+i_{t-1}u_2)P_{T-t-2}X_{T-t}\nonumber \\ {}&\qquad +(h_{t-1}u_2\rho _2)P_{T-t-2}X_{T-t-1} +(g_{t-1}\gamma +m_{t-1}\rho _1+n_{t-1})W_{T-t}X_{T-t}\nonumber \\ {}&\qquad +(m_{t-1}\rho _2)W_{T-t}X_{T-t-1}+(2d_{t-1}\rho _1\rho _2+h_{t-1}\gamma \rho _2\nonumber \\&\qquad +o_{t-1}\rho _2)X_{T-t}X_{T-t-1}+(a_{t-1}\sigma ^2_{\epsilon }+d_{t-1}\sigma ^2_{\eta }+p_{t-1})\Big ] \end{aligned}$$
(58)
Thus at time period \(T-t\) the unrestricted minimization of (58) occurs at following point:
$$\begin{aligned} S_{T-t}= & {} \Big (\frac{g_{t-1}u_1+j_{t-1}-u_1-f_{t-1}\theta -2a_{t-1}u_1\theta }{2(\theta +a_{t-1}\theta ^2+c_{t-1}-g_{t-1}\theta )}\Big )P_{T-t-1}\nonumber \\&+\Big (\frac{g_{t-1}u_2-2a_{t-1}u_2\theta -u_2}{2(\theta +a_{t-1}\theta ^2+c_{t-1}-g_{t-1}\theta )}\Big )P_{T-t-2}\nonumber \\&+\Big (\frac{2c_{t-1}-g_{t-1}\theta }{2(\theta +a_{t-1}\theta ^2+c_{t-1}-g_{t-1}\theta )}\Big )W_{T-t}\nonumber \\&+\Big (\frac{n_{t-1}+m_{t-1}\rho _1-i_{t-1}\theta +g_{t-1}\gamma -2a_{t-1}\theta \gamma -\gamma -h_{t-1}\theta \rho _1}{2(\theta +a_{t-1}\theta ^2+c_{t-1}-g_{t-1}\theta )}\Big )X_{T-t}\nonumber \\&+\Big (\frac{m_{t-1}\rho _2-h_{t-1}\theta \rho _1}{2(\theta +a_{t-1}\theta ^2+c_{t-1}-g_{t-1}\theta )}\Big )X_{T-t-1} \nonumber \\= & {} \delta _{P,t}P_{T-t-1}+\delta _{PP,t}P_{T-t-2}+\delta _{W,t}W_{T-t}+\delta _{X,t}X_{T-t}+\delta _{XX,t}X_{T-t-1}. \nonumber \\ \end{aligned}$$
(59)
Putting point of minima \(S_{T-t}\) from (59) into (58), we get
$$\begin{aligned} V_{T-t}= & {} a_tP_{T-t-1}^2+b_tP_{T-t-2}^2+c_tW_{T-t}^2+d_tX_{T-t}^2+e_tX_{T-t-1}^2+f_tP_{T-t-1}P_{T-t-2}\nonumber \\&+g_tP_{T-t-1}W_{T-t} +h_tP_{T-t-1}X_{T-t}+i_tP_{T-t-1}X_{T-t-1}+j_tP_{T-t-2} W_{T-t}\nonumber \\&+k_tP_{T-t-2}X_{T-t}+l_tP_{T-t-2}X_{T-t-1} +m_tW_{T-t}X_{T-t}+n_tW_{T-t}X_{T-t-1}\nonumber \\ {}&+o_tX_{T-t}X_{T-t-1}+p_t. \end{aligned}$$
(60)
In view of (55)–(60), Proposition 1 is proved. \(\square \)
Appendix 2
In this Appendix, the proof of Lemma 1 is presented.
Proof
First we prove the Eq. (13) of Lemma 1.
Consider the case when t is an even positive integer.
That is let \(t=2m\) for some positive integer m.
$$\begin{aligned} u_1L_t+u_2L_{t-1}= & {} u_1L_{2m}+u_2L_{2m-1}\\= & {} u_1\Big [\displaystyle \sum ^{m}_{k=0}{2m-k\atopwithdelims ()k}u_1^{2m-2k}u_2^k\Big ]+u_2\Big [\displaystyle \sum ^{m-1}_{k=0}{2m-1-k\atopwithdelims ()k}u_1^{2m-1-2k}u_2^k\Big ]\\= & {} \Big [\displaystyle \sum ^{m}_{k=0}{2m-k\atopwithdelims ()k}u_1^{2m+1-2k}u_2^k\Big ]+\Big [\displaystyle \sum ^{m-1}_{k=0}{2m-1-k\atopwithdelims ()k}u_1^{2m-1-2k}u_2^{k+1}\Big ]\\= & {} \Big [\displaystyle \sum ^{m}_{k=0}{2m-k\atopwithdelims ()k}u_1^{2m+1-2k}u_2^k\Big ]+\Big [\displaystyle \sum ^{m}_{k=1}{2m-k\atopwithdelims ()k-1}u_1^{2m+1-2k}u_2^{k}\Big ]\\= & {} \Big [u_1^{2m+1}+\displaystyle \sum ^{m}_{k=1}{2m-k\atopwithdelims ()k}u_1^{2m+1-2k}u_2^k\Big ]+\Big [\displaystyle \sum ^{m}_{k=1}{2m-k\atopwithdelims ()k-1}u_1^{2m+1-2k}u_2^{k}\Big ]\\= & {} u_1^{2m+1}+\displaystyle \sum ^{m}_{k=1}\Big [{2m-k\atopwithdelims ()k}+{2m-k\atopwithdelims ()k-1}\Big ]u_1^{2m+1-2k}u_2^k\\= & {} u_1^{2m+1}+\displaystyle \sum ^{m}_{k=1}{2m+1-k\atopwithdelims ()k}u_1^{2m+1-2k}u_2^k\\= & {} \displaystyle \sum ^{m}_{k=0}{2m+1-k\atopwithdelims ()k}u_1^{2m+1-2k}u_2^k\\= & {} L_{2m+1}\\= & {} L_{t+1} \end{aligned}$$
Similarly Eq. (13) can be proved for the case when t is an odd positive integer.
Similar to the above proof, the proof of Eq. (14) follows. \(\square \)
Appendix 3
The proof of Lemma 2 is given in this Appendix C.
Proof
We will use method of mathematical induction to prove the Eq. (15) of Lemma 2.
Consider
$$\begin{aligned} X_2= & {} \rho _1X_1+\rho _2X_{0}+\eta _2\\= & {} X_1(M_1)+\rho _2X_{0}(M_0)+\displaystyle \sum _{k=0}^{2-2}M_{k}\eta _{2-k}. \end{aligned}$$
Thus Eq. (15) is true for \(t=2\).
Let Eq. (15) be true for all t such that \(2\le t\le m\) for some positive integer m where \(2<m<T\).
To prove Lemma 2, next we need to show that Eq. (15) is true for \(t=m+1\).
$$\begin{aligned} X_{m+1}= & {} \rho _1X_m+\rho _2X_{m-1}+\eta _{m+1}\\= & {} \rho _1\Big [X_1(M_{m-1})+\rho _2X_{0}(M_{m-2})+\displaystyle \sum _{k=0}^{m-2}M_{k}\eta _{m-k}\Big ] \\&+\rho _2\Big [X_1(M_{m-2})+\rho _2X_{0}(M_{m-3})+\displaystyle \sum _{k=0}^{m-3}M_{k}\eta _{m-1-k}\Big ]+\eta _{m+1}\\= & {} X_1(\rho _1M_{m-1}+\rho _2M_{m-2})+\rho _2X_{0}(\rho _1M_{m-2}+\rho _2M_{m-3})\\&+\Big [(\rho _1M_0)\eta _m+\displaystyle \sum _{k=1}^{m-2}(\rho _1M_k+\rho _2M_{k-1})\eta _{m-k}\Big ]+\eta _{m+1}\\&(\hbox {By Eq. (14) of Lemma 1 and by equation}~~\rho _1M_0=M_1)\\= & {} X_1(M_{m})+\rho _2X_{0}(M_{m-1})+\Big [M_1\eta _m+\displaystyle \sum _{k=1}^{m-2}M_{k+1}\eta _{m-k}\Big ]+\eta _{m+1}\\= & {} X_1(M_{m})+\rho _2X_{0}(M_{m-1})+\displaystyle \sum _{k=0}^{m-1}M_{k}\eta _{m+1-k} \end{aligned}$$
Thus claim is proved. \(\square \)
Appendix 4
In this Appendix, proof of Lemma 3 is presented.
Proof
The Eq. (16) of Lemma 3 is obvious.
We will use method of mathematical induction to prove the Eq. (17) of Lemma 3.
Consider
$$\begin{aligned} P_2= & {} u_1P_1+u_2P_{0}+\theta {\hat{S}}_2+\gamma X_2+\epsilon _2\\&(\hbox {Putting values of }P_2\hbox { and }X_2\hbox { from Eqs. (3) and (4), respectively, we get})\\= & {} u_1(u_1P_0+u_2P_{-1}+\theta {\hat{S}}_1+\gamma X_1+\epsilon _1)+u_2P_{0}+\theta {\hat{S}}_2+\gamma (\rho _1X_1+\rho _2X_{0}+\eta _2)+\epsilon _2\\= & {} P_0(L_2)+u_2P_{-1}(L_1)+\gamma X_1(L_0M_1+L_1M_0)+\gamma \rho _2X_{0}(L_0M_0)\\&+(L_0\epsilon _2+L_1\epsilon _1)+\gamma (L_0M_0\eta _2) \end{aligned}$$
Thus Eq. (17) is true for \(t=2\).
Let Eq. (17) be true for all t such that \(2\le t\le m\) for some positive integer m where \(2<m<T\).
To prove Lemma 3, next we need to show that Eq. (17) is true for \(t=m+1\).
$$\begin{aligned} P_{m+1}= & {} u_1P_m+u_2P_{m-1}+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1}\\= & {} u_1\left[ P_0(L_m)+u_2P_{-1}(L_{m-1})+\gamma X_1\left( \displaystyle \sum _{k=0}^{m-1}L_kM_{m-1-k}\right) +\gamma \rho _2X_{0}\left( \displaystyle \sum _{k=0}^{m-2}L_kM_{m-2-k}\right) \right. \\&\left. +\theta \left( \displaystyle \sum _{k=0}^{m-1}L_k{\hat{S}}_{m-k}\right) +\left( \displaystyle \sum _{k=0}^{m-1}L_k\epsilon _{m-k}\right) + \gamma \left\{ \displaystyle \sum _{k=0}^{m-2}\left( \displaystyle \sum _{l=0}^{k}L_lM_{k-l}\right) \eta _{m-k}\right\} \right] \\&+u_2\left[ P_0(L_{m-1})+u_2P_{-1}(L_{m-2})+\gamma X_1\left( \displaystyle \sum _{k=0}^{m-2}L_kM_{m-2-k}\right) +\gamma \rho _2X_{0}\left( \displaystyle \sum _{k=0}^{m-3}L_kM_{m-3-k}\right) \right. \\&\left. +\theta \left( \displaystyle \sum _{k=0}^{m-2}L_k{\hat{S}}_{m-1-k}\right) +\left( \displaystyle \sum _{k=0}^{m-2}L_k\epsilon _{m-1-k}\right) + \gamma \left\{ \displaystyle \sum _{k=0}^{m-3}\left( \displaystyle \sum _{l=0}^{k}L_lM_{k-l}\right) \eta _{m-1-k}\right\} \right] \\&+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1}\\= & {} P_0(u_1L_m+u_2L_{m-1})+u_2P_{-1}(u_1L_{m-1}+u_2L_{m-2})+\gamma X_1\left\{ (u_1L_0)M_{m-1}\right. \\&\left. +\displaystyle \sum _{k=1}^{m-1}(u_1L_k+u_2L_{k-1})M_{m-1-k}\right\} +\gamma \rho _2 X_{0}\left\{ (u_1L_0)M_{m-2}\right. \\&\left. +\displaystyle \sum _{k=1}^{m-2}(u_1L_k+u_2L_{k-1})M_{m-2-k}\right\} +\theta \left\{ (u_1L_0){\hat{S}}_m+\displaystyle \sum _{k=1}^{m-1} (u_1L_k+u_2L_{k-1}){\hat{S}}_{m-k}\right\} \\&+\left\{ (u_1L_0)\epsilon _m+\displaystyle \sum _{k=1}^{m-1}(u_1L_k+u_2L_{k-1})\epsilon _{m-k}\right\} \\ \end{aligned}$$
$$\begin{aligned}&\quad \quad +\gamma \left[ (u_1L_0)M_0\eta _m +\displaystyle \sum _{k=1}^{m-2}\left\{ (u_1L_0)M_k+\displaystyle \sum _{l=1}^{k}(u_1L_l+u_2L_{l-1})M_{k-l}\right\} \eta _{m-k}\right] +\theta {\hat{S}}_{m+1}\\&\quad \quad +\gamma X_{m+1}+\epsilon _{m+1}\\&\quad (\hbox {By using Eq. (13) of Lemma 1 and the equation}~~u_1L_0=L_1.)\\= & {} P_0(L_{m+1})+u_2P_{-1}(L_{m})+\gamma X_1\left\{ L_1M_{m-1}+\displaystyle \sum _{k=1}^{m-1}L_{k+1}M_{m-1-k}\right\} +\gamma \rho _2 X_{0}\left\{ L_1M_{m-2}\right. \\&\left. +\displaystyle \sum _{k=1}^{m-2}L_{k+1}M_{m-2-k}\right\} +\theta \left\{ L_1{\hat{S}}_m+\displaystyle \sum _{k=1}^{m-1}L_{k+1}{\hat{S}}_{m-k}\right\} +\left\{ L_1\epsilon _m+\displaystyle \sum _{k=1}^{m-1}L_{k+1}\epsilon _{m-k}\right\} \\&+\gamma \left[ (L_1M_0)\eta _m+\displaystyle \sum _{k=1}^{m-2}\left\{ L_1M_k+\displaystyle \sum _{l=1}^{k}L_{l+1}M_{k-l}\right\} \eta _{m-k}\right] +\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1}\\&(\hbox {Simplifying above equation further and by using Eq. (15) of Lemma 2.})\\= & {} P_0(L_{m+1})+u_2P_{-1}(L_{m})+\gamma X_1\left\{ \displaystyle \sum _{k=1}^{m}L_{k}M_{m-k}\right\} +\gamma \rho _2 X_{0}\left\{ \displaystyle \sum _{k=1}^{m-1}L_{k}M_{m-1-k}\right\} \\&+\theta \left\{ \displaystyle \sum _{k=1}^{m}L_{k}{\hat{S}}_{m+1-k}\right\} +\left\{ \displaystyle \sum _{k=1}^{m}L_{k}\epsilon _{m+1-k}\right\} +\gamma \left[ \displaystyle \sum _{k=0}^{m-2} \left\{ \displaystyle \sum _{l=0}^{k}L_{l+1}M_{k-l}\right\} \eta _{m-k}\right] \\&+\theta {\hat{S}}_{m+1}+\gamma \left[ X_1(M_{m})+\rho _2X_{0}(M_{m-1})+\displaystyle \sum _{k=0}^{m-1}M_{k}\eta _{m+1-k}\right] +\epsilon _{m+1}\\= & {} P_0(L_{m+1})+u_2P_{-1}(L_{m})+\gamma X_1\left\{ \displaystyle \sum _{k=0}^{m}L_{k}M_{m-k}\right\} +\gamma \rho _2 X_{0}\left\{ \displaystyle \sum _{k=0}^{m-1}L_{k}M_{m-1-k}\right\} \\&+\theta \left\{ \displaystyle \sum _{k=0}^{m}L_{k}{\hat{S}}_{m+1-k}\right\} +\left\{ \displaystyle \sum _{k=0}^{m}L_{k}\epsilon _{m+1-k}\right\} +\gamma \left[ \displaystyle \sum _{k=0}^{m-1} \left\{ \displaystyle \sum _{l=0}^{k}L_{l}M_{k-l}\right\} \eta _{m+1-k}\right] \end{aligned}$$
Thus Lemma 3 is proved. \(\square \)
Appendix 5
The proof of Lemma 4 is presented in this Appendix.
Proof
We use method of mathematical induction to prove Eq. (23) of Lemma 4.
Consider
$$\begin{aligned} Cov_1(X_3,X_2)= & {} Cov_1(\rho _1 X_2+\rho _2 X_1+\eta _3,X_2) \\= & {} \rho _1 Var_1(X_2). \end{aligned}$$
Thus Eq. (23) is true for \(t=3\).
Let Eq. (23) be true for all t such that \(3\le t\le m\) for some positive integer m where \(3<m<T\).
To prove Lemma 4, next we need to show that Eq. (23) is true for \(t=m+1\).
$$\begin{aligned}&Cov_1(X_{m+1},X_{m})\\&\quad =Cov_1(\rho _1 X_m+\rho _2 X_{m-1}+\eta _{m+1},X_m) \\&\quad = \rho _1Var_1(X_m)+\rho _2Cov_1(X_m,X_{m-1})\\&\qquad \hbox {(By the assumption that Eq. (23) is true for all} t \hbox { such that } 3\le t\le m)\\&\quad = \rho _1Var_1(X_m)+\rho _2[\rho _1\{Var_1(X_{m-1})+\rho _2Var_1(X_{m-2})+\cdots +\rho _2^{m-3}Var_1(X_2)\}]\\&\quad = \rho _1[Var_1(X_m)+\rho _2Var_1(X_{m-1})+\rho _2^2Var_1(X_{m-2})+\cdots +\rho _2^{m-2}Var_1(X_2)] \end{aligned}$$
Thus Lemma 4 is proved. \(\square \)
Appendix 6
The proof of Lemma 5 is presented in this Appendix.
Proof
We consider the Eq. (24) of Lemma 5 as the statement corresponding to t. With this we prove (24) by the method of mathematical induction.
First we will show that Eq. (24) is true for \(t=4\).
For \(t=4\), we have \(2\le k\le 4-2\), that is \(k=2\).
$$\begin{aligned} Cov_1(X_4,X_{4-2})= & {} Cov_1(X_4,X_2)\\= & {} Cov_1(\rho _1X_3+\rho _2X_2+\eta _4,X_2)\\= & {} \rho _1Cov_1(X_3,X_2)+\rho _2Cov_1(X_2,X_2)\\= & {} M_1Cov_1(X_3,X_2)+\rho _2M_0Var_1(X_2) \end{aligned}$$
Assume that Eq. (24) is true for \(4\le t\le m\) for some positive integer m where \(4<m<T\).
In particular, we have following equations.
For \(t=m\) and corresponding \(2\le k\le m-2\)
$$\begin{aligned} Cov_1(X_{m},X_{m-k})=M_{k-1}Cov_1(X_{m-k+1},X_{m-k})+\rho _2M_{k-2}Var_1(X_{m-k}) \end{aligned}$$
(61)
For \(t=m-1\) and corresponding \(2\le k\le m-3\)
$$\begin{aligned} Cov_1(X_{m-1},X_{m-1-k})=M_{k-1}Cov_1(X_{m-k},X_{m-1-k})+\rho _2M_{k-2}Var_1(X_{m-1-k}). \end{aligned}$$
(62)
To prove the Lemma 5, next we need to show that Eq. (24) is true for \(t=m+1\).
That is we need to show that \(\forall 2\le k\le m-1\).
$$\begin{aligned} Cov_1(X_{m+1},X_{m+1-k})=M_{k-1}Cov_1(X_{m+1-k+1},X_{m+1-k})+\rho _2M_{k-2}Var_1(X_{m+1-k}). \end{aligned}$$
(63)
We prove Eq. (63) separately for \(k=2\), \(k=3\) and for \(4\le k\le m-1\).
For \(k=2\), consider
$$\begin{aligned} Cov_1(X_{m+1},X_{m+1-2})= & {} Cov_1(X_{m+1},X_{m-1}) \nonumber \\= & {} Cov_1(\rho _1X_{m}+\rho _2X_{m-1}+\eta _{m+1},X_{m-2}) \nonumber \\= & {} \rho _1Cov_1(X_m,X_{m-1})+\rho _2Cov_1(X_{m-1},X_{m-1}) \nonumber \\= & {} M_1Cov_1(X_m,X_{m-1})+\rho _2M_0Var_1(X_{m-1}). \end{aligned}$$
(64)
For \(k=3\), consider
$$\begin{aligned}&Cov_1(X_{m+1},X_{m+1-3})\nonumber \\&\quad =Cov_1(X_{m+1},X_{m-2}) \nonumber \\&\quad =Cov_1(\rho _1X_{m}+\rho _2X_{m-1}+\eta _{m+1},X_{m-2}) \nonumber \\&\quad =\rho _1Cov_1(X_m,X_{m-2})+\rho _2Cov_1(X_{m-1},X_{m-2}) \nonumber \\&\quad =\rho _1Cov_1(\rho _1X_{m-1}+\rho _2X_{m-2}+\eta _m,X_{m-2})+\rho _2Cov_1(X_{m-1},X_{m-2}) \nonumber \\&\quad =\rho _1[\rho _1Cov_1(X_{m-1},X_{m-2})+\rho _2Cov_1(X_{m-2},X_{m-2})]+\rho _2Cov_1(X_{m-1},X_{m-2}) \nonumber \\&\quad =(\rho _1^2+\rho _2)Cov_1(X_{m-1},X_{m-2})+\rho _2\rho _1Cov_1(X_{m-2},X_{m-2}) \nonumber \\&\quad =M_2Cov_1(X_{m-1},X_{m-2})+\rho _2M_1Var_1(X_{m-2}). \end{aligned}$$
(65)
For \(4\le k\le m-1\), we consider
$$\begin{aligned} Cov_1(X_{m+1},X_{m+1-k})= & {} Cov_1(\rho _1X_m+\rho _2X_{m-1}+\eta _{m+1},X_{m+1-k}) \nonumber \\= & {} \rho _1Cov_1(X_m,X_{m+1-k})+\rho _2Cov_1(X_{m-1},X_{m+1-k})\nonumber \\ \end{aligned}$$
(66)
We evaluate terms \(Cov_1(X_m,X_{m+1-k})\) and \(Cov_1(X_{m-1},X_{m+1-k})\) of above Eq. (66) as follows:
$$\begin{aligned}&Cov_1(X_m,X_{m+1-k})\nonumber \\&\quad =Cov_1(X_m,X_{m-(k-1)}) \nonumber \\&\qquad (\hbox {Let }k^{'}=k-1.\hbox { Thus }4\le k\le m-1~~\Rightarrow ~~3\le k^{'} \le m-2.) \nonumber \\&\quad = Cov_1(X_m,X_{m-k^{'}}) \nonumber \\&\qquad (\hbox {Applying Eq. (61), in particular for } 3\le k^{'}\le m-2\hbox {, we get}) \nonumber \\&\quad =M_{k^{'}-1}Cov_1(X_{m-k^{'}+1},X_{m-k^{'}})+\rho _2M_{k^{'}-2}Var_1(X_{m-k^{'}}) \nonumber \\&\qquad (\hbox {Put }k^{'}=k-1.) \nonumber \\&\quad =M_{k-2}Cov_1(X_{m-k+2},X_{m-k+1})+\rho _2M_{k-3}Var_1(X_{m-k+1}) \end{aligned}$$
(67)
$$\begin{aligned}&\quad \qquad Cov_1(X_{m-1},X_{m+1-k})=Cov_1(X_{m-1},X_{m-1-(k-2)}) \nonumber \\&\qquad (\hbox {Let }k^{'}=k-2.\hbox { Thus }4\le k\le m-1~~\Rightarrow ~~2 \le k^{'} \le m-3.) \nonumber \\&\quad = Cov_1(X_{m-1},X_{m-1-k^{'}}) \nonumber \\&\qquad (\hbox {Applying Eq. (62)}) \nonumber \\&\quad = M_{k^{'}-1}Cov_1(X_{m-1-k^{'}+1},X_{m-1-k^{'}})+\rho _2M_{k^{'}-2}Var_1(X_{m-1-k^{'}}) \nonumber \\&\quad = M_{k^{'}-1}Cov_1(X_{m-k^{'}},X_{m-k^{'}-1})+\rho _2M_{k^{'}-2}Var_1(X_{m-k^{'}-1}) \nonumber \\&\qquad (\hbox {Put }k^{'}=k-2.) \nonumber \\&\quad =M_{k-3}Cov_1(X_{m-k+2},X_{m-k+1})+\rho _2M_{k-4}Var_1(X_{m-k+1}) \end{aligned}$$
(68)
Using Eqs. (67) and (68) in Eq. (66), we get \(\forall 4\le k\le m-1\)
$$\begin{aligned}&Cov_1(X_{m+1},X_{m+1-k})\nonumber \\&\quad = \rho _1[M_{k-2}Cov_1(X_{m-k+2},X_{m-k+1})+\rho _2M_{k-3}Var_1(X_{m-k+1})]+ \nonumber \\&\qquad \rho _2[M_{k-3}Cov_1(X_{m-k+2},X_{m-k+1})+\rho _2M_{k-4}Var_1(X_{m-k+1})] \nonumber \\&\quad =(\rho _1M_{k-2}+\rho _2M_{k-3})Cov_1(X_{m-k+2},X_{m-k+1})\nonumber \\&\qquad +\rho _2(\rho _1M_{k-3}+\rho _2M_{k-4})Var_1(X_{m-k+1}) \nonumber \\&\qquad (\hbox {By Eq. (14) of Lemma 1}) \nonumber \\&\quad = M_{k-1}Cov_1(X_{m-k+2},X_{m-k+1})+\rho _2M_{k-2}Var_1(X_{m-k+1}) \nonumber \\&\quad =M_{k-1}Cov_1(X_{m+1-k+1},X_{m+1-k})+\rho _2M_{k-2}Var_1(X_{m+1-k}) \end{aligned}$$
(69)
In view of (64), (65) and (69), the Eq. (63) is proved.
Thus Lemma 5 is proved. \(\square \)
Appendix 7
The proof of Lemma 6 is presented in this Appendix.
Proof
We consider the Eq. (30) of Lemma 6 as the statement corresponding to t. With this we prove the Eq. (30) by the method of mathematical induction.
Consider
$$\begin{aligned} Cov_1(P_2,X_k)= & {} Cov_1(u_1P_1+u_2P_{0}+\theta {\hat{S}}_2+\gamma X_2+\epsilon _2,X_k)\\= & {} \gamma Cov_1(X_2,X_k)\\= & {} \gamma [L_0Cov_1(X_2,X_k)]. \end{aligned}$$
Thus Eq. (30) is true for \(t=2\).
Let Eq. (30) be true for all t such that \(2\le t\le m\) for some positive integer m where \(2<m<T\).
To prove the Lemma 6, next we need to show that Eq. (30) is true for \(t=m+1\).
$$\begin{aligned}&Cov_1(P_{m+1},X_k)\nonumber \\&\quad = Cov_1(u_1P_{m}+u_2P_{m-1}+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1},X_k)\\&\quad =u_1Cov_1(P_m,X_k)+u_2Cov_1(P_{m-1},X_k)+\gamma Cov_1(X_{m+1},X_k)\\&\qquad (\hbox {By the assumption that Eq. (30) is true for all }t\hbox { such that }2\le t\le m)\\&\quad =u_1\gamma [L_0Cov_1(X_m,X_k)+L_1Cov_1(X_{m-1},X_k)+\cdots +L_{m-2}Cov_1(X_2,X_k)]+ \\&\qquad u_2\gamma [L_0Cov_1(X_{m-1},X_k)+L_1Cov_1(X_{m-2},X_k)+\cdots +L_{m-3}Cov_1(X_2,X_k)]\\&\qquad +\gamma Cov_1(X_{m+1},X_k)\\&\quad = \gamma [(u_1L_0)Cov_1(X_m,X_k)+(u_1L_1+u_2L_0)Cov_1(X_{m-1},X_k)\\&\qquad +(u_1L_2+u_2L_1)Cov_1(X_{m-2},X_k)+\cdots +(u_1L_{m-2}+u_2L_{m-3})Cov_1(X_2,X_k)]\\&\qquad +\gamma Cov_1(X_{m+1},X_k)\\&\qquad \hbox {(By using Eq. (13) of Lemma 1 and the equation } u_1L_0=L_1)\\&\quad =\gamma [L_0Cov_1(X_{m+1},X_k)+L_1Cov_1(X_{m},X_k)+\cdots +L_{m-1}Cov_1(X_2,X_k)] \end{aligned}$$
Thus Lemma 6 is proved. \(\square \)
Appendix 8
In this Appendix, we present the proof of Lemma 7.
Proof
We will use method of mathematical induction to prove the Eq. (31) of Lemma 7.
Consider
$$\begin{aligned} Cov_1(P_2,P_1)= & {} Cov_1(u_1 P_1+u_2 P_{0}+\theta {\hat{S}}_2+\gamma X_2+\epsilon _2,P_1) \\= & {} u_1Var_1(P_1)+\gamma Cov_1(X_2,P_1) \\= & {} u_1Var_1(P_1)+\gamma Cov_1(P_1,X_2). \end{aligned}$$
Thus Eq. (31) is true for \(t=2\).
Let Eq. (31) be true for all n such that \(2\le t\le m\) for some positive integer m where \(2<m<N\).
To prove the Lemma 5, next we need to show that Eq. (31) is true for \(t=m+1\).
$$\begin{aligned}&Cov_1(P_{m+1},P_{m})\\&\quad =Cov_1(u_1 P_m+u_2 P_{m-1}+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1},P_m) \\&\quad = u_1Var_1(P_m)+u_2Cov_1(P_{m-1},P_m)+\gamma Cov_1(X_{m+1},P_m)\\&\quad = u_1Var_1(P_m)+u_2Cov_1(P_m,P_{m-1})+\gamma Cov_1(X_{m+1},P_m)\\&\qquad \hbox {(By the assumption that Eq. (31) is true for all }n \hbox { such that }2\le n\le m)\\&\quad = u_1Var_1(P_m)+u_2\Big [u_1\Big \{Var_1(P_{m-1})+u_2Var_1(P_{m-2})+\cdots +u_2^{m-2}Var_1(P_1)\Big \}\\&\qquad +\gamma \Big \{Cov_1(P_{m-1},X_{m})+u_2Cov_1(P_{m-2},X_{m-1})+\cdots +u_2^{m-2}Cov_1(P_{1},X_{2})\Big \}\Big ]\\&\qquad +\gamma Cov_1(X_{m+1},P_m)\\&\quad = u_1[Var_1(P_m)+u_2Var_1(P_{m-1})+u_2^2Var_1(P_{m-2})+\cdots +u_2^{m-1}Var_1(P_1)]\\&\qquad +\gamma [Cov_1(P_{m},X_{m+1})+u_2Cov_1(P_{m-1},X_{m})+\cdots +u_2^{m-1}Cov_1(P_{1},X_{2})] \end{aligned}$$
Thus Lemma 7 is proved. \(\square \)
Appendix 9
In this Appendix, proof of Lemma 8 is presented.
Proof
We consider the Eq. (32) of Lemma 8 as the statement corresponding to n. With this we prove the Eq. (32) by the method of mathematical induction.
First we will show that Eq. (32) is true for \(n=3\).
For \(n=3\), we have \(2\le k\le 3-1\), that is \(k=2\).
$$\begin{aligned} Cov_1(P_3,P_{3-2})= & {} Cov_1(P_3,P_1)\\= & {} Cov_1(u_1P_2+u_2P_1+\theta {\hat{S}}_3+\gamma X_3+\epsilon _3,P_1)\\= & {} u_1Cov_1(P_2,P_1)+u_2Cov_1(P_1,P_1)+\gamma Cov_1(X_3,P_1)\\= & {} L_1Cov_1(P_2,P_1)+u_2L_0Var_1(P_1)+\gamma [L_0 Cov_1(P_1,X_3)] \end{aligned}$$
Assume that Eq. (32) is true for \(3\le n\le m\) for some positive integer m where \(3<m<N\). In particular we have following equations.
For \(n=m\) and corresponding \(2\le k\le m-1,\)
$$\begin{aligned} Cov_1(P_{m},P_{m-k})= & {} L_{k-1}Cov_1(P_{m-k+1},P_{m-k})+u_2L_{k-2}Var_1(P_{m-k}) \nonumber \\&\gamma [L_0Cov_1(P_{m-k},X_m)+L_1Cov_1(P_{m-k},X_{m-1})+\cdots \nonumber \\&+L_{k-2}Cov_1(P_{m-k},X_{m-k+2})]. \end{aligned}$$
(70)
For \(n=m-1\) and corresponding \(2\le k\le m-2,\)
$$\begin{aligned} Cov_1(P_{m-1},P_{m-1-k})= & {} L_{k-1}Cov_1(P_{m-k},P_{m-1-k})+u_2L_{k-2}Var_1(P_{m-1-k}) \nonumber \\&+\gamma [L_0Cov_1(P_{m-1-k},X_{m-1}) +L_1Cov_1(P_{m-1-k},X_{m-2})\nonumber \\&+\cdots +L_{k-2}Cov_1(P_{m-1-k},X_{m-k+1})]. \end{aligned}$$
(71)
To prove the Lemma 8, next we need to show that Eq. (32) is true for \(n=m+1\).
That is we need to show that \(\forall 2\le k\le m\).
$$\begin{aligned} Cov_1(P_{m+1},P_{m+1-k})= & {} L_{k-1}Cov_1(P_{m+1-k+1},P_{m+1-k})+u_2L_{k-2}Var_1(P_{m+1-k}) \nonumber \\&+\gamma [L_0Cov_1(P_{m+1-k},X_{m+1})\nonumber \\&+L_1Cov_1(P_{m+1-k},X_{m})\nonumber \\&+\cdots +L_{k-2}Cov_1(P_{m+1-k},X_{m-k+3})] \end{aligned}$$
(72)
We prove the Eq. (72) separately for \(k=2\), \(k=3\) and for \(4\le k\le m\).
For \(k=2\), we consider
$$\begin{aligned} Cov_1(P_{m+1},P_{m+1-2})= & {} Cov_1(P_{m+1},P_{m-1}) \nonumber \\= & {} Cov_1(u_1P_m+u_2P_{m-1}+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1},P_{m-1}) \nonumber \\= & {} u_1Cov_1(P_m,P_{m-1})+u_2Cov_1(P_{m-1},P_{m-1})\nonumber \\&+\gamma Cov_1(X_{m+1},P_{m-1}) \nonumber \\= & {} L_1Cov_1(P_m,P_{m-1})+u_2L_0Var_1(P_{m-1})\nonumber \\&+\gamma [L_0 Cov_1(P_{m-1},X_{m+1})] \end{aligned}$$
(73)
For \(k=3\), we consider
$$\begin{aligned}&Cov_1(P_{m+1},P_{m+1-3})\nonumber \\&\quad =Cov_1(P_{m+1},P_{m-2}) \nonumber \\&\quad =Cov_1(u_1P_m+u_2P_{m-1}+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1},P_{m-2}) \nonumber \\&\quad =u_1Cov_1(P_m,P_{m-2})+u_2Cov_1(P_{m-1},P_{m-2})+\gamma Cov_1(X_{m+1},P_{m-2}) \nonumber \\&\quad =u_1[u_1Cov_1(P_{m-1},P_{m-2})+u_2Cov_1(P_{m-2},P_{m-2})+\gamma Cov_1(X_{m},P_{m-2})] \nonumber \\&\qquad +u_2Cov_1(P_{m-1},P_{m-2})+\gamma Cov_1(X_{m+1},P_{m-2}) \nonumber \\&\quad =(u_1^2+u_2)Cov_1(P_{m-1},P_{m-2})+u_2u_1Cov_1(P_{m-2},P_{m-2}) \nonumber \\&\qquad \gamma [Cov_1(X_{m+1},P_{m-2})+u_1Cov_1(X_{m},P_{m-2})] \nonumber \\&\quad =L_2Cov_1(P_{m-1},P_{m-2})+u_2L_1Var_1(P_{m-2}) \nonumber \\&\qquad \gamma [L_0Cov_1(P_{m-2},X_{m+1})+L_1Cov_1(P_{m-2},X_{m})]. \end{aligned}$$
(74)
For \(4\le k\le m\), we consider
$$\begin{aligned} Cov_1(P_{m+1},P_{m+1-k})= & {} Cov_1(u_1P_m+u_2P_{m-1}+\theta {\hat{S}}_{m+1}+\gamma X_{m+1}+\epsilon _{m+1},P_{m+1-k}) \nonumber \\= & {} u_1Cov_1(P_m,P_{m+1-k})+u_2Cov_1(P_{m-1},P_{m+1-k}) \nonumber \\&+\gamma Cov_1(X_{m+1},P_{m+1-k}) ~~~~~~ \end{aligned}$$
(75)
We evaluate terms \(Cov_1(P_m,P_{m+1-k})\) and \(Cov_1(P_{m-1},P_{m+1-k})\) of above Eq. (75) as follows.
$$\begin{aligned} Cov_1(P_m,P_{m+1-k})= & {} Cov_1(P_m,P_{m-(k-1)}) \nonumber \\&(\hbox {Let }k^{'}=k-1.\hbox { Thus }4\le k\le m~~\Rightarrow ~~3\le k^{'} \le m-1.) \nonumber \\= & {} Cov_1(P_m,P_{m-k^{'}}) \nonumber \\&(\hbox {Applying Eq. (70), in particular for }3\le k^{'}\le m-1,\hbox { we get}) \nonumber \\= & {} L_{k^{'}-1}Cov_1(P_{m-k^{'}+1},P_{m-k^{'}})+u_2L_{k^{'}-2}Var_1(P_{m-k^{'}}) \nonumber \\&+\gamma [L_0Cov_1(P_{m-k^{'}},X_m)+L_1Cov_1(P_{m-k^{'}},X_{m-1})+\cdots \nonumber \\&+L_{k^{'}-2}Cov_1(P_{m-k^{'}},X_{m-k^{'}+2})] \nonumber \\ {}&(\hbox {Put }k^{'}=k-1.) \nonumber \\= & {} L_{k-2}Cov_1(P_{m-k+2},P_{m-k+1})+u_2L_{k-3}Var_1(P_{m-k+1}) \nonumber \\ {}&+\gamma [L_0Cov_1(P_{m-k+1},X_m) +L_1Cov_1(P_{m-k+1},X_{m-1})\nonumber \\ {}&+\cdots +L_{k-3}Cov_1(P_{m-k+1},X_{m-k+3})] \end{aligned}$$
(76)
$$\begin{aligned} Cov_1(P_{m-1},P_{m+1-k})= & {} Cov_1(P_{m-1},P_{m-1-(k-2)}) \nonumber \\&(\hbox {Let }k^{'}=k-2.\hbox { Thus }4\le k\le m~~\Rightarrow ~~2\le k^{'} \le m-2.) \nonumber \\= & {} Cov_1(P_{m-1},P_{m-1-k^{'}}) \nonumber \\&(\hbox {Applying Eq. (71)}) \nonumber \\= & {} L_{k^{'}-1}Cov_1(P_{m-k^{'}},P_{m-1-k^{'}})+u_2L_{k^{'}-2}Var_1(P_{m-1-k^{'}}) \nonumber \\ {}&+\gamma [L_0Cov_1(P_{m-1-k^{'}},X_{m-1}) +L_1Cov_1(P_{m-1-k^{'}},X_{m-2})\nonumber \\ {}&+\cdots +L_{k^{'}-2}Cov_1(P_{m-1-k^{'}},X_{m-k^{'}+1})] \nonumber \\&(\hbox {Put }k^{'}=k-2.) \nonumber \\= & {} L_{k-3}Cov_1(P_{m-k+2},P_{m-k+1})+u_2L_{k-4}Var_1(P_{m-k+1}) \nonumber \\ {}&+\gamma [L_0Cov_1(P_{m-k+1},X_{m-1}) +L_1Cov_1(P_{m-k+1},X_{m-2})\nonumber \\ {}&+\cdots +L_{k-4}Cov_1(P_{m-k+1},X_{m-k+3})] \end{aligned}$$
(77)
Using Eqs. (76) and (77) in Eq. (75), we get \(\forall 4\le k\le m\)
$$\begin{aligned} Cov_1(P_{m+1},P_{m+1-k})= & {} u_1\Big \{L_{k-2}Cov_1(P_{m-k+2},P_{m-k+1})+u_2L_{k-3}Var_1(P_{m-k+1}) \nonumber \\ {}&+\gamma [L_0Cov_1(P_{m-k+1},X_m) +L_1Cov_1(P_{m-k+1},X_{m-1})+\cdots \nonumber \\ {}&+L_{k-3}Cov_1(P_{m-k+1},X_{m-k+3})]\Big \}\nonumber \\&+u_2\Big \{L_{k-3}Cov_1(P_{m-k+2},P_{m-k+1})+u_2L_{k-4}Var_1(P_{m-k+1}) \nonumber \\&+\gamma [L_0Cov_1(P_{m-k+1},X_{m-1}) +L_1Cov_1(P_{m-k+1},X_{m-2})+\cdots \nonumber \\&+L_{k-4}Cov_1(P_{m-k+1},X_{m-k+3})]\Big \} +\gamma Cov_1(X_{m+1},P_{m+1-k}) \nonumber \\= & {} (u_1L_{k-2}+u_2L_{k-3})Cov_1(P_{m-k+2},P_{m-k+1})+u_2(u_1L_{k-3}\nonumber \\ {}&+u_2L_{k-4})Var_1(P_{m-k+1}) +\gamma [u_1L_0Cov_1(P_{m-k+1},X_m)\nonumber \\ {}&+(u_1L_1+u_2L_0)Cov_1(P_{m-k+1},X_{m-1})+\cdots \nonumber \\ {}&+(u_1L_{k-3}+u_2L_{k-4})Cov_1(P_{m-k+1},X_{m-k+3})] \nonumber \\ {}&+\gamma Cov_1(X_{m+1},P_{m+1-k}) \nonumber \\&(\hbox {By Eq. (13) Lemma 1 and the equation } u_1L_0=L_1) \nonumber \\= & {} L_{k-1}Cov_1(P_{m-k+2},P_{m-k+1})+u_2L_{k-2}Var_1(P_{m-k+1}) \nonumber \\ {}&+\gamma [L_0Cov_1(X_{m+1},P_{m+1-k}) \nonumber \\&+L_1Cov_1(P_{m-k+1},X_m)+L_2Cov_1(P_{m-k+1},X_{m-1})+\cdots \nonumber \\&+L_{k-2}Cov_1(P_{m-k+1},X_{m-k+3})] \nonumber \\= & {} L_{k-1}Cov_1(P_{m+1-k+1},P_{m+1-k})+u_2L_{k-2}Var_1(P_{m+1-k}) \nonumber \\ {}&+\gamma [L_0Cov_1(P_{m+1-k},X_{m+1})\nonumber \\&+L_1Cov_1(P_{m+1-k},X_{m})+\cdots +L_{k-2}Cov_1(P_{m+1-k},X_{m-k+3})] \nonumber \\ \end{aligned}$$
(78)
In view of (73), (74) and (78), the Eq. (72) is proved.
Thus Lemma 8 is proved. \(\square \)
