Skewness and kurtosis of sum of lognormals Below is an equation $f(x)$ that is the sum of two lognormal waveforms.  All of the coefficients are known, and by integration, the area under the curve is simply $P_1 + P_2$.  
$\displaystyle f(x) = \frac {P_1 } {S_1(x + D) \sqrt { 2 \Pi}} e ^ {-(\ln(x + D)-M_1) ^2  / 2S_1^2}  +  \frac {P_2 } {S_2(x + D) \sqrt { 2 \Pi}} e ^ {-(\ln(x + D)-M_2) ^2  / 2S_2^2} + B$
Treating these two waveforms as probability distributions, the skewness and kurtosis of the individual waveforms can be calculated from the coefficients $S_1$ and $S_2$ as
Skewness = 
$(e^{S_x^2}\!\!+2) \sqrt{e^{S_x^2}\!\!-1}$
Kurtosis = 
$\displaystyle e ^ {4S_x^2} + 2e ^ {3S_x^2} + 3e ^ {2S_x^2} - 6$
If $f(x)$ itself represents a probability distribution, how can one obtain the skewness and kurtosis of $f(x)$ from these same coefficients?


Long story short:  I worked out the solution. 
Disregarding the measurement baseline B and shared delay D above, we have the following pdf: $$ f_X(x) = \frac {1} {P_1 + P_2} \left ( \frac {P_1 } {S_1x \sqrt { 2 \pi}} e ^ {-(\ln(x)-M_1) ^2 / (2S_1^2)} + \frac {P_2 } {S_2x \sqrt { 2 \pi}} e ^ {-(\ln(x)-M_2) ^2 / (2S_2^2)} \right ) $$ 
Using the same substitution method given here for the single lognormal pdf, the mgf of the dual lognormal pdf can be calculated as:
$$ {\rm E}[X^k] = \frac {P_1} {P_1 + P_2} e^{k(2M_1 + kS_1^2)/2} + \frac {P_2} {P_1 + P_2} e^{k(2M_2 + kS_2^2)/2} $$ 
Once, the mgf is known, the moments can be trivially calculated, noting $ \mu_2 = \sigma^2 $, γ1 = $ \mu_3 / \sigma^3 $ and κ1, is equal to $ \mu_4 / \sigma^4 $.
