{
  "$schema": "../../../schemas/mko-v0.5.schema.json",
  "schema_version": "mko-v0.5",
  "id": "mko-weak-law-large-numbers",
  "semantic_family_id": "sf-weak-law-large-numbers",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "弱大數律"
  },
  "summary": {
    "zh-Hant": "IID且具有有限variance時，Chebyshev界給樣本平均依機率收斂到共同mean。"
  },
  "statement": {
    "zh-Hant": "限明示IID通用證書、共同有限mean mu與variance sigma squared。independence先給Var(sample mean)=sigma squared/n，再用exact Chebyshev epsilon-delta tail證依機率收斂；有限模擬不足。"
  },
  "formula": {
    "tex": "\\bar{X}_n\\xrightarrow{\\mathrm{P}}\\mu",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mover accent=\"true\"><mi>X</mi><mo>¯</mo></mover><mi>n</mi></msub><mover><mo>→</mo><mi mathvariant=\"normal\">P</mi></mover><mi>μ</mi></mrow></math>",
    "semantic_ast": {
      "type": "tends_to_probability",
      "expression": {
        "type": "subscript",
        "base": {
          "type": "overline",
          "body": {
            "type": "symbol",
            "name": "X",
            "glyph": "X"
          }
        },
        "subscript": {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      },
      "target": {
        "type": "symbol",
        "name": "mu",
        "glyph": "μ"
      }
    },
    "compiler": {
      "id": "ocme-formula-source13",
      "version": "0.5.0",
      "source_sha256": "0ffd8d8857ddd0886155cf4eeb638b78fa11f75c85f815dce50f328e8e9ac3cd"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "IID是全序列任意有限子族的symbolic certificate，不是有限樣本觀察。"
    },
    {
      "id": "a2",
      "zh-Hant": "共同mean與variance有限，epsilon與delta皆為正有理數。"
    }
  ],
  "symbols": [
    {
      "token": "\\bar{X}_n",
      "role": "sample_mean",
      "role_zh": "樣本平均",
      "scope": "representative_formula"
    },
    {
      "token": "\\mu",
      "role": "common_mean",
      "role_zh": "共同期望",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "iid_sequence",
      "role_zh": "同分布獨立隨機變數列",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "positive_sample_size",
      "role_zh": "正整數樣本數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-iid-sequence",
      "title_zh": "獨立同分布序列",
      "reason_zh": "independence與共同law。"
    },
    {
      "id": "mko-expectation",
      "title_zh": "有限模型的期望值",
      "reason_zh": "共同mean。"
    },
    {
      "id": "mko-variance",
      "title_zh": "有限機率模型的變異數",
      "reason_zh": "有限variance與平均variance。"
    },
    {
      "id": "mko-convergence-in-probability",
      "title_zh": "依機率收斂",
      "reason_zh": "epsilon tail目標。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "independence把平均的variance縮成sigma squared/n。",
      "Chebyshev界隨n單調趨零並給全稱tail witness。",
      "結論只有依機率收斂，不是a.s.、CLT或有限n等於極限。",
      "記號對齊候選：bar(X)_n=(X_1+...+X_n)/n；μ 是共同期望，箭頭上直立 P 明示依機率收斂。保留全序列 IID 與有限變異數的既有前提；不是單一 X_n 的收斂、幾乎必然收斂或有限模擬證明。"
    ],
    "common_misconception_zh": "由幾次模擬或一段樣本路徑宣稱大數律已證，或把弱大數律當CLT。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪採有限variance的Chebyshev版本；不宣稱最一般WLLN或strong law。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入strong law與central limit theorem的獨立前提。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-weak-law-large-numbers.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-11-05 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}