Gamma Convergence For Beginners Pdf 11 [PATCHED]

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This was a bit surprising to me. Even though I knew the $\chi^2$ distribution -- a distribution of the sum of squared standard normal RVs -- was a special case of the gamma, I didn't realise the gamma was essentially just a generalisation allowing for the sum of normal random variables of any variance. This also leads to other characterisations I had not come across before, such as the exponential distribution being equivalent to the sum of two squared normal distributions.

This is all somewhat mysterious to me. Is the normal distribution fundamental to the derivation of the gamma distribution, in the manner I outlined above? Most resources I checked make no mention that the two distributions are intrinsically related like this, or even for that matter describe how the gamma is derived. This makes me think some lower-level truth is at play that I have simply highlighted in a convoluted way?

Let us address the question posed, This is all somewhat mysterious to me. Is the normal distribution fundamental to the derivation of the gamma distribution...? No mystery really, it is simply that the normal distribution and the gamma distribution are members, among others of the exponential family of distributions, which family is defined by the ability to convert between equational forms by substitution of parameters and/or variables. As a consequence, there are many conversions by substitution between distributions, a few of which are summarized in the figure below.

First A more direct relationship between the gamma distribution (GD) and the normal distribution (ND) with mean zero follows. Simply put, the GD becomes normal in shape as its shape parameter is allowed to increase. Proving that that is the case is more difficult. For the GD, $$\text{GD}(z;a,b)=\begin{array}{cc}& \begin{cases} \dfrac{b^{-a} z^{a-1} e^{-\dfrac{z}{b}}}{\Gamma (a)} & z>0 \\ 0 & \text{other} \\\end{cases}\,. \\\end{array}$$

Second Let us make the point that due to the similarity of form between these distributions, one can pretty much develop relationships between the gamma and normal distributions by pulling them out of thin air. To wit, we next develop an "unfolded" gamma distribution generalization of a normal distribution.

Note first that it is the semi-infinite support of the gamma distribution that impedes a more direct relationship with the normal distribution. However, that impediment can be removed when considering the half-normal distribution, which also has a semi-infinite support. Thus, one can generalize the normal distribution (ND) by first folding it to be half-normal (HND), relating that to the generalized gamma distribution (GD), then for our tour de force, we "unfold" both (HND and GD) to make a generalized ND (a GND), thusly.

$$\text{GD}\left(x;\alpha ,\beta ,\gamma ,\mu \right)=\begin{array}{cc} & \begin{cases} \dfrac{\gamma e^{-\left(\dfrac{x-\mu }{\beta }\right)^{\gamma }} \left(\dfrac{x-\mu }{\beta }\right)^{\alpha \gamma -1}}{\beta \,\Gamma (\alpha )} & x>\mu \\ 0 & \text{other} \\\end{cases} \\\end{array}\,,$$

As Alecos Papadopoulos already noted there is no deeper connection that makes sums of squared normal variables 'a good model for waiting time'. The gamma distribution is the distribution for a sum of generalized normal distributed variables. That is how the two come together.

But the type of sum and type of variables may be different. While the gamma distribution, when derived from the exponential distribution (p=1), gets the interpretation of the exponential distribution (waiting time), you can not go reverse and go back to a sum of squared Gaussian variables and use that same interpretation.

@article{DalMaso1987,author = {Dal Maso, Gianni},journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},keywords = {relaxed problems; Choquet capacity; -convergence},language = {eng},number = {3},pages = {423-464},publisher = {Scuola normale superiore},title = {$\Gamma $-convergence and $\mu $-capacities},url = { },volume = {14},year = {1987},}

TY - JOURAU - Dal Maso, GianniTI - $\Gamma $-convergence and $\mu $-capacitiesJO - Annali della Scuola Normale Superiore di Pisa - Classe di ScienzePY - 1987PB - Scuola normale superioreVL - 14IS - 3SP - 423EP - 464LA - engKW - relaxed problems; Choquet capacity; -convergenceUR - ER - 2b1af7f3a8