It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. For those interested in the theory, the fact that cases 1, 2 and 3 yield convergence to the Gaussian distribution is a consequence of the Central Limit Theorem under the Liapounov condition. More specifically, and because the samples produced here come from uniformly bounded distributions (we use a random number generator to simulate uniform deviates), all that is needed for convergence to the Gaussian distribution is that the sum of the squares of the weights -- and thus Stdev(S) as n tends to infinity -- must be infinite. More generally, we can work with more complex auto-regressive processes with a covariance matrix as general as possible, then compute S as a weighted sum of the X(k)'s, and find a relationship between the weights and the covariance matrix, to eventually identify conditions on the covariance matrix that guarantee convergence to the Gaussian destribution.

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. For those interested in the theory, the fact that cases 1, 2 and 3 yield convergence to the Gaussian distribution is a consequence of the Central Limit Theorem under the Liapounov condition. More specifically, and because the samples produced here come from uniformly bounded distributions (we use a random number generator to simulate uniform deviates), all that is needed for convergence to the Gaussian distribution is that the sum of the squares of the weights -- and thus Stdev(S) as n tends to infinity -- must be infinite. More generally, we can work with more complex auto-regressive processes with a covariance matrix as general as possible, then compute S as a weighted sum of the X(k)'s, and find a relationship between the weights and the covariance matrix, to eventually identify conditions on the covariance matrix that guarantee convergence to the Gaussian destribution.

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. Case 1: a(k) 1, corresponding to the classic version of the Central Limit Theorem, and with guaranteed convergence to the Gaussian distribution. Case 2: a(k) 1 / log 2k, still with guaranteed convergence to the Gaussian distribution Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.) Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.)

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. Case 1: a(k) 1, corresponding to the classic version of the Central Limit Theorem, and with guaranteed convergence to the Gaussian distribution. Case 2: a(k) 1 / log 2k, still with guaranteed convergence to the Gaussian distribution Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.) Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.)

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. Case 1: a(k) 1, corresponding to the classic version of the Central Limit Theorem, and with guaranteed convergence to the Gaussian distribution. Case 2: a(k) 1 / log 2k, still with guaranteed convergence to the Gaussian distribution Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.) Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.)

It turned out that putting more weight on close neighbors, and increasingly lower weight on far away neighbors (with weights slowly decaying to zero based on the distance to the neighbor in question) was the solution to the problem. By normalization, I mean considering (S - E(S)) / Stdev(S) instead of S. Produce n 10,000 random deviates X(1),...,X(n) uniformly distrubuted on [0, 1] Compute S based on a specific set of weights a(1),...,a(n) Compute S based on a specific set of weights a(1),...,a(n) Each of the above m iterations provides one value of the limiting distribution. Case 2: a(k) 1 / log 2k, still with guaranteed convergence to the Gaussian distribution Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.) Case 3: a(k) k {-1/2}, the last exponent (-1/2) that still provides guaranteed convergence to the Gaussian distribution, according to the Central Limit Theorem with the Liapounov condition (more on this below.)