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A minor thought I found inspiring many years ago: the Central Limit Theorem may be considered a statement about dynamic processes in the functional space of distributions. If you consider the operator of convolution (rescaling for unit variance), the normal distribution is the only attractor (under many "natural" choices of metrics).
Mandatory link to a 3blue1brown video: https://m.youtube.com/watch?v=KuXjwB4LzSA For me, the way to understanding convolutions was to to put sample rand(0, 1) a bunch of times and bucket the samples, then plot the number of samples in a bucket. You get a more-or-less flat graph. Now if you add or multiply two variables, you get different shapes. Once you spent enough effort on trying to understand why, you derive con…
Mandatory link to a 3blue1brown video: https://m.youtube.com/watch?v=KuXjwB4LzSA For me, the way to understanding convolutions was to to put sample rand(0, 1) a bunch of times and bucket the samples, then plot the number of samples in a bucket. You get a more-or-less flat graph. Now if you add or multiply two variables, you get different shapes. Once you spent enough effort on trying to understand why, you derive con…
the (discretized) probability density the (discrete) signal intensity the coefficients of the polynomial
If we look at the discrete case, since the operation of convolution is computationally expensive (O(n^3)), people often rely on the Fast Fourier Transform Algorithm (FFT), that runs in O(n log(n)). An excellent and very enjoyable video on the FFT applied to the polynomial multiplication is https://youtu.be/h7apO7q16V0?t=1.
A minor thought I found inspiring many years ago: the Central Limit Theorem may be considered a statement about dynamic processes in the functional space of distributions. If you consider the operator of convolution (rescaling for unit variance), the normal distribution is the only attractor (under many "natural" choices of metrics).
Can you elaborate / add reference?
As the author mentions, convolutions for signal processing and convolutions for probability are essentially equivalent things in two different contexts. It is interesting that the same concept is behind the multiplication of polynomials. In the discrete case we have the parallel of the vectors representing ... the (discretized) probability density the (discrete) signal intensity the coefficients of the polynomial ...…
A minor thought I found inspiring many years ago: the Central Limit Theorem may be considered a statement about dynamic processes in the functional space of distributions. If you consider the operator of convolution (rescaling for unit variance), the normal distribution is the only attractor (under many "natural" choices of metrics).
In the generating function pic, the distribution is borne as coefficients of polynomials or of formal power series, and convolution is polynomial multiplication. One example of your step is multiplying by (x+1)/2 (Bernoulli trial), and that gives approaches to the gaussian by chunks of normalized binomial coefficients.
Other related limit, described as a functional central limit theorem is given by Donsker's theorem [1], giving the passage from the discrete situation (random walk) to the continous (Brownian motion).
Nice jazz pun.
Earlier quoted context omitted.
Can you elaborate / add reference?
"Central Limit Theorem" says that "adding (bounded variance) stuff together makes it converge to a Gaussian" (roughly). On the other hand, when you are adding random variable together, you are actually convolving their densities. Thus, what the CLT says is that a gaussian is some sort of attractor/fixed point of the "dynamic process" of convolving (finite energy/bounded variance) distributions.