2.6 Uniqueness and Continuity
Everything above extracts information from a generating function. Two theorems in the other direction are what make the tool legitimate, and neither is usually stated explicitly — although the proof of the central limit theorem later in these notes depends on the second of them entirely.
Theorem 2.6.1 (Uniqueness). If two non-negative integer-valued random variables \(X\) and \(Y\) satisfy \(G_X(t) = G_Y(t)\) for all \(t\) in some interval containing the origin, then \(X\) and \(Y\) have the same distribution.
Proof. A power series that converges on an interval about the origin determines its coefficients uniquely, by repeated differentiation at \(0\). Applying \(P(X=n) = G_X^{(n)}(0)/n!\) to each series gives \(P(X=n)=P(Y=n)\) for every \(n\). □
Note. This is what licenses the standard argument “the generating function is that of a Poisson distribution, therefore the variable is Poisson”. Without uniqueness the conclusion would not follow, and that argument is used repeatedly from here on.
Theorem 2.6.2 (Continuity). Let \(X_1, X_2, \dots \) be a sequence of random variables and \(X\) another, all non-negative and integer-valued. If \(G_{X_n}(t) \rightarrow G_X(t)\) for every \(t\in (0,1)\), then \(X_n \underset {D}{\longrightarrow } X\).
Remark. The continuity theorem is the bridge between generating functions and limits: convergence of the functions gives convergence of the distributions. Its proof belongs to analysis and is not given here, but its role should be clear. The central limit theorem is proved later by showing that a moment generating function converges to \(e^{t^{2}/2}\); without a continuity theorem that calculation would say nothing at all about the distributions themselves.
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