The following expected value is part of a limit I'm trying to evaluation (for the Lyapunov CLT). My question is essentially is this a valid approach, I have the expected value of the absolute value of a function and the absolute value is cubed.

$$E\big[ \mid d_{i}w_{i} - \phi w_{i} \mid^{3}\big] $$ where $d_{i} \sim Bern(\phi)$, and $w_{i}$ is a weight for each of the Bernoulli random variables that can take values between $[\frac{1}{m}, 1]$ where $m$ is the number of of random variables. To simplify the limit I need to calculate I have proceeded as follows:

$$E\big[ \mid d_{i}w_{i} - \phi w_{i} \mid^{3}\big] = E\big[ (w_{i} (d_{i} - \phi))^{3} \mid w_{i} (d_{i} - \phi) >0 \big] \cdot P(w_{i} (d_{i} - \phi)>0) - \\E\big[ (w_{i} (d_{i} - \phi))^{3} \mid w_{i} (d_{i} - \phi) < 0\big] \cdot P(w_{i} (d_{i} - \phi)<0) $$ and am then able to solve this expected value and solve the limit showing the random variable in questions satisfies the Lyapunov CLT. Is this a legitimate way to evaluate this expected value?


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