Chapter Summary

Chapter Summary

Key Points

• 1.

  The MGF $M_X(t) = \mathbb{E}[e^{tX}]$ encodes all moments. When it exists in a neighborhood of the origin, it determines the distribution uniquely and yields moments via differentiation: $\mathbb{E}[X^k] = M_X^{(k)}(0)$. Its key limitation is that it may not exist for heavy-tailed distributions.

• 2.

  The CF $\phi_X(u) = \mathbb{E}[e^{juX}]$ always exists. It is bounded by $1$, uniformly continuous, Hermitian, and non-negative definite. It uniquely determines the distribution via the Fourier inversion formula. The Levy continuity theorem connects convergence of CFs to convergence in distribution.

• 3.

  The PGF $G_X(s) = \mathbb{E}[s^X]$ is tailored to counting. For nonneg. integer-valued RVs, it encodes the PMF as Taylor coefficients. Factorial moments are obtained by evaluating derivatives at $s = 1$. The compounding theorem $G_S(s) = G_N(G_X(s))$ handles random sums elegantly.

• 4.

  All transforms convert convolution to multiplication. If $X \perp Y$, then $\phi_{X+Y} = \phi_X \cdot \phi_Y$ (and similarly for MGF and PGF). This is the fundamental algebraic property that makes transform methods powerful for analyzing sums of independent RVs.

• 5.

  The LLN and CLT follow from CF convergence. The LLN: $\phi_{S_n/n}(u) \to e^{j\mu u}$. The CLT: $\phi_{(S_n - n\mu)/(\sigma\sqrt{n})}(u) \to e^{-u^2/2}$. Both proofs use Taylor expansion plus the Levy continuity theorem.

• 6.

  Cumulants are additive for independent sums. The CGF $m_X(t) = \logM_X(t)$ generates cumulants. The Gaussian has $\kappa_n = 0$ for $n \geq 3$, so the CLT can be understood as the vanishing of higher cumulants in the normalized sum.

• 7.

  Cramer's theorem gives the exact exponential decay rate. $\mathbb{P}(S_n > na) \doteq e^{-n m_X^*(a)}$ where $m_X^*$ is the Fenchel-Legendre transform of the CGF. The Chernoff bound provides the upper bound; the tilted distribution technique gives the matching lower bound.

• 8.

  Branching process extinction is determined by $\mu = G'(1)$. The extinction probability $\eta$ is the smallest non-negative solution of $s = G(s)$. If $\mu \leq 1$: $\eta = 1$ (certain extinction). If $\mu > 1$: $\eta < 1$ (survival possible).