## 3 Results  
### DfG and Economic Growth  
There are two types of distributions for the RCA index. One is the distribution of the RCA index for all the sectors/products of an economy or a country. The other is the distribution of the RCA for all countries in the world given a sector/product. In this article, we focus on the former. Since the RCA index is the ratio of two probabilities, it is useful to first understand the distributions of the two probabilities. It turns out that both the numerator and the denominator defining the RCA index (\(p_{(ix)}^{k}\) and \(p_{(ux)}^{k}\)) basically follow exponentially truncated Zipf-Mandelbrot's law, given by:  
\[p(k)\sim(k+p)^{-\alpha}e^{-\beta k^{\gamma}},k>k^{*}, \tag{11}\]  
where \(p\), \(\alpha\), \(\beta\), and \(\gamma\) are parameters. The exponential truncation can be naturally expected due to finiteness of the data. To better understand deviations from normality in the distribution of RCA index for different countries, we use Japan and Germany as two examples. Figure 1 shows the distribution features of the two parts of RCA index and the probability distribution function (PDF) of the RCA index for Japan and Germany under the HS4 scheme in 2018. Obviously, the \(p_{(ux)}\) in Figure 1a,b follows exponentially truncated Zipf-Mandelbrot's law. If the \(p_{(ix)}\) in Figure 1a,b are also arranged in descending order, they will also follow exponentially truncated Zipf-Mandelbrot's law (but possibly with different parameters). Interestingly, by comparing the layout of \(p_{(ix)}\) (red diamonds) around \(p_{(ux)}\) (blue circles) in Figure 1a,b, we can observe that the \(p_{(ix)}\) of Germany is more concentrated around \(p_{(ux)}\) than Japan's. This highlights that Germany's export share of most products relative to its total exports is closer to the world average level than Japan's. Next, we discuss how the differences between Figure 1a,b results in the differences in the distribution of the RCA index shown Figure 1c,d. Clearly, the PDFs for the RCA index of Germany and Japan are very different. Concretely, the PDF of Japan's RCA index has more asymmetry, stronger skewness, and longer tail than that of Germany's. This suggests that the PDF of Germany's RCA index should be closer to a Gaussian distribution than Japan's. To better quantify how the PDF of a country's RCA index deviates from normality, we employ DfG we have defined earlier. The DfG for Germany and Japan is 0.088 and 0.266 in 2018, respectively. According to the nature of DfG--the more positive DfG, the larger the deviation from Gaussianity, one can conclude that the PDF of Germany's RCA index is indeed closer to a normal distribution than that of Japan's, just as one has anticipated from Figure 1.  
It is interesting to examine the spatiotemporal evolution of the DfG of all the economies in the world. For this purpose, we have systematically computed DfG for all the economies in the world from 1991 to 2019. The spatial variations of the DfG in 1998, 2008 and 2018 are illustrated in Figure 2. We observe that the variations of DfG are characterized by spatiotemporal heterogeneity and regional spatial agglomeration. First, let us focus on the spatiotemporal heterogeneity. From Figure 2a, we can observe: (1) only the DfG of USA and Germany was less than 0.1, followed by France and Italy, (2) only a few countries (such as China, South Korea, Japan, etc.) had DfG between 0.2 and 0.3, and (3) the DfG of most countries was greater than 0.3, especially in Africa, South America, Southern and Western Asia, and Eastern Europe. By 2008, which is shown in Figure 2b, the spatial variation of DfG had undergone some changes. Now Germany is the only economy with DfG \(<\) 0.1, indicating that Germany is the only country with the PDF of its RCA index to be very close to a normal distribution. The decrease in China's DfG was significant. In contrast, the DfG in some countries has become larger, such as USA, France, Australia, Egypt, etc.