Sample means converge in distribution to normal. Lindeberg–Lévy (i.i.d.), Lyapunov (non-identically distributed). Foundation for asymptotic inference.
graph TD
D1["Def: Convergence in\ndistribution Xₙ↝X"]
D2["Def: i.i.d. sequence"]
D3["Def: Standard normal\nΦ distribution"]
T1["Thm: Lindeberg–Lévy\n√n(Sₙ/n−μ)/σ ↝ N(0,1)\ni.i.d. finite variance"]
T2["Thm: Lyapunov CLT\nnon-identical, Lyapunov cond"]
T3["Thm: Berry–Esseen\nrate of convergence"]
T4["Thm: Multivariate CLT"]
L1["Lemma: Char functions\nφₓₙ(t)→φₓ(t)"]
T5["Thm: Delta method\nθ̂ asymptotically normal"]
D1 --> T1
D2 --> T1
D3 --> T1
D1 --> T2
T1 --> T3
T1 --> T4
T1 --> L1
T1 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 15
- Edges: 11
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: High-dimensional CLT, Stein's method, rate of convergence (quantitative CLT). math.PR