Graduate How to Derive a Closed Form for a Double Sum with Stochastic Variables?

[tworitdash]

I have a equation with a double sum. However, one of the variables in one of the sums comes from a stochastic distribution (Gaussian to be specific). How can I get a closed form equivalent of this expression? The U and Tare constants in the equation.

$$ \sum_{k = 0}^{N_k-1} \bigg [ \big[ \sum_{i}^{N_i} \cos(\frac{4\pi}{\lambda} u_i k T) - \cos(\frac{4\pi}{\lambda} U k T) \big]^2 + \big[ \sum_{i = 1}^{N_i} \sin(\frac{4\pi}{\lambda} u_i k T) - \sin(\frac{4\pi}{\lambda} U k T) \big]^2 \bigg]$$

$$ u_i \thicksim \mathcal{N}(\mu_{u}, \sigma_{u}) $$

 

[anuttarasammyak]

I do not see what is $u_i$ as a function of number i.

 

[tworitdash]

I do not see what is $u_i$ as a function of number i.

The values of $u_i$ come from a Gaussian Distribution I explain in the second equation. It is a random sample drawn from the same distribution.

 

[anuttarasammyak]

Expecting that average including sin is zero, the formula would become
N_i^2 \sum_k <\cos u, k>^2 - 2N_i \sum_k <\cos u,k> (\cos U,k)+ N_k
approximately for large N_i under random sampling for Gaussian distribution where
$<\cos u, k>$ is average of the cos u function given for i samplings with given constant k.
with $<sin, k>=0$

 

[Delta2]

Wondering what sort of problem gave you this monster sums. Let me guess something about the velocities of molecules of a gas at temperature T and internal energy U?

 

[tworitdash]

Wondering what sort of problem gave you this monster sums. Let me guess something about the velocities of molecules of a gas at temperature T and internal energy U?

You almost got it. These are particle velocities #u_i#, but the other parameters are related to motion rather than temperature. #T# is the time step, #k# is the time index. This is sort of a likelihood (DFT) equation where the cyclic velocity (frequency) is #U#. It actually comes from a complex exponential expression, where I have separated the real and the imaginary parts (cos and sin) to have a better function for likelihood. #N_i# are the number of particles and #N_k# are the number of time steps.