Let the random variables 
where 
Once again, we let the sequence of decompositions be 
Next, we show that
![\displaystyle E(\theta_{k+1}|\mathcal{D}_{k}) \\ = E\left[\sum_{i=1}^{k}\eta_{i+1}f_{i}(\eta_{1},\dots,\eta_{i})|\mathcal{D}_{k}\right] \\ = \sum_{i=1}^{k}E\left[\eta_{i+1}f_{i}(\eta_{1},\dots,\eta_{i})|\mathcal{D}_{k}\right] \\ = \sum_{i=1}^{k}E\left[\eta_{i+1}|\mathcal{D}_{k}\right]f_{i}(\eta_{1},\dots,\eta_{i})\mbox{ since }f_{i}(\eta_{1},\dots,\eta_{i})\mbox{ is }\mathcal{D}_{k\ge i}\mbox{-measurable} \\ = E\left[\eta_{k+1}|\mathcal{D}_{k}\right]f_{i}(\eta_{1},\dots,\eta_{k})+\sum_{i=1}^{k-1}E\left[\eta_{i}|\mathcal{D}_{k}\right]f_{i}(\eta_{1},\dots,\eta_{i}) \\ = \sum_{i=1}^{k-1}E\left[\eta_{i}|\mathcal{D}_{k}\right]f_{i}(\eta_{1},\dots,\eta_{i})\mbox{ since we are given }E\left[n_{k}|\eta_{1},\dots,\eta_{k-1}\right]=0 \\ = \sum_{i=1}^{k-1}\eta_{i+1}f_{i}(\eta_{1},\dots,\eta_{i})\mbox{ since }\eta_{i}\mbox{ is }\mathcal{D}_{k\ge i}\mbox{-measurable} \\ = \theta_{k}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++E%28%5Ctheta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%29%09%5C%5C+%3D+E%5Cleft%5B%5Csum_%7Bi%3D1%7D%5E%7Bk%7D%5Ceta_%7Bi%2B1%7Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D+%5C%5C+%3D+%5Csum_%7Bi%3D1%7D%5E%7Bk%7DE%5Cleft%5B%5Ceta_%7Bi%2B1%7Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5D+%5C%5C+%3D+%5Csum_%7Bi%3D1%7D%5E%7Bk%7DE%5Cleft%5B%5Ceta_%7Bi%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29%5Cmbox%7B+since+%7Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29%5Cmbox%7B+is+%7D%5Cmathcal%7BD%7D_%7Bk%5Cge+i%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+E%5Cleft%5B%5Ceta_%7Bk%2B1%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bk%7D%29%2B%5Csum_%7Bi%3D1%7D%5E%7Bk-1%7DE%5Cleft%5B%5Ceta_%7Bi%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29+%5C%5C+%3D+%5Csum_%7Bi%3D1%7D%5E%7Bk-1%7DE%5Cleft%5B%5Ceta_%7Bi%7D%7C%5Cmathcal%7BD%7D_%7Bk%7D%5Cright%5Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29%5Cmbox%7B+since+we+are+given+%7DE%5Cleft%5Bn_%7Bk%7D%7C%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bk-1%7D%5Cright%5D%3D0+%5C%5C+%3D+%5Csum_%7Bi%3D1%7D%5E%7Bk-1%7D%5Ceta_%7Bi%2B1%7Df_%7Bi%7D%28%5Ceta_%7B1%7D%2C%5Cdots%2C%5Ceta_%7Bi%7D%29%5Cmbox%7B+since+%7D%5Ceta_%7Bi%7D%5Cmbox%7B+is+%7D%5Cmathcal%7BD%7D_%7Bk%5Cge+i%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+%5Ctheta_%7Bk%7D+&bg=ffffff&fg=333333&s=0&c=20201002)
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Martingales – Problem (28/365)
Let the random variables
where
Once again, we let the sequence of decompositions be
Next, we show that
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