{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-convergence-in-probability",
  "semantic_family_id": "sf-convergence-in-probability",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "依機率收斂"
  },
  "summary": {
    "zh-Hant": "X_n依機率收斂到X，若對每個epsilon>0，偏差超過epsilon的機率趨近0。"
  },
  "statement": {
    "zh-Hant": "全部X_n與X須在同一機率空間，並保留forall epsilon>0、forall delta>0、exists N、forall n>=N的tail責任。本輪使用明示c/(n+s) deviation-probability profile或eventual equality證書；有限prefix不構成收斂。"
  },
  "formula": {
    "tex": "P(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "82bda20a28179ce1d5f24b4cb9405369004e46c8a2f65647e3c93bbe3f8fa6db"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "symbolic profile c/(n+s)使用正整數c,s並保持機率不超1的起始域。"
    },
    {
      "id": "a2",
      "zh-Hant": "不混同almost sure、L^p或distribution convergence。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "convergence_in_probability",
      "role_zh": "依機率收斂紀錄",
      "scope": "representative_formula"
    },
    {
      "token": "X",
      "role": "random_variable",
      "role_zh": "共同極限變數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-joint-distribution",
      "title_zh": "有限聯合分布",
      "reason_zh": "共同有限機率空間。"
    },
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "X_n與X為同空間變數。"
    },
    {
      "id": "mko-probability",
      "title_zh": "機率",
      "reason_zh": "deviation事件機率。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "變數依n索引。"
    },
    {
      "id": "mko-metric-convergence",
      "title_zh": "度量收斂",
      "reason_zh": "實值偏差使用距離。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每個正epsilon都要控制deviation event。",
      "c/(n+s)給任意delta的明示N。",
      "有限prefix只能audit，eventual equality提供另一全稱證書。"
    ],
    "common_misconception_zh": "看到有限多個error probabilities下降就宣稱收斂，或把依分布收斂當同一概念。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只接受明示symbolic tail或eventual equality證書；不推出a.s.、Lp或CLT。",
    "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": "加入依分布收斂與弱大數律。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-convergence-in-probability.md 整理；2026-10-14 脈衝 R07（使用者手動接續 2026-10-30 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"
  }
}