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I wonder if there is a Zipf's Law equivalent of the Glivenko-Cantelli Theorem, where the condition of independence is replaced by requiring a sum of powers remain fixed, and the identical assumption is replaced by a marginal identical assumption?

If $C=\sum_{i=1}^n X_i^\alpha$ and $X_i \sim P(X)$ then $\frac{1}{n} \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} I(X_i \leq x) \propto \ln(x)$ almost surely?

I ask this because empirically Zipf's Law seems to occur when the sample violates independence, and is usually constrained by requiring a fixed sum (like fixed population size, wealth, or resource).

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I wonder if there is a Zipf's Law equivalent of the Glivenko-Cantelli Theorem, where the condition of independence is replaced by requiring a sum of powers remain fixed, and the identical assumption is replaced by a marginal identical assumption?

If $C=\sum_{i=1}^n X_i^\alpha$ and $X_i \sim P(X)$ then $\frac{1}{n} \sum_{i=1}^{n} I(X_i \leq x) \propto \ln(x)$

I ask this because empirically Zipf's Law seems to occur when the sample violates independence, and is usually constrained by requiring a fixed sum (like fixed population size, wealth, or resource).