Sample average converges to expectation. Weak LLN (convergence in probability); Strong LLN (almost sure). Foundation for frequentist inference.
graph TD
D1["Def: Expectation E(X)\n∫ X dP"]
D2["Def: Conv in probability\nXₙ →ᵖ X"]
D3["Def: Almost sure conv\nXₙ → X a.s."]
T1["Thm: Weak LLN\nSₙ/n →ᵖ μ\ni.i.d. finite mean"]
T2["Thm: Strong LLN\nKolmogorov\nSₙ/n → μ a.s.\ni.i.d. finite var"]
T3["Thm: Etemadi LLN\nSₙ/n → μ a.s.\nequal marginal dist"]
T4["Thm: Ergodic theorem\nBirkhoff: time avg → space avg"]
L1["Lemma: Borel–Cantelli"]
D1 --> T1
D1 --> T2
D2 --> T1
D3 --> T2
D1 --> T3
D3 --> T3
T2 --> T4
L1 --> T2
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 theorem
class L1 lemma
Process Statistics
- Nodes: 14
- Edges: 10
- Definitions: 3
- Theorems: 4
- Lemmas: 1
Frontier: Non-stationary sequences, concentration inequalities (Hoeffding, Azuma), empirical process theory. math.PR