{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-convergence-in-distribution",
  "semantic_family_id": "sf-convergence-in-distribution",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "依分布收斂"
  },
  "summary": {
    "zh-Hant": "極限CDF每個連續點上F_n逐點收斂到F時，稱依分布收斂。"
  },
  "statement": {
    "zh-Hant": "不要求共同機率空間或coupling。有限prefix不證收斂，極限CDF不連續點不要求收斂；依分布收斂不一般推出依機率、a.s.或Lp。"
  },
  "formula": {
    "tex": "D(X)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>D</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "D",
        "glyph": "D"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "X",
          "glyph": "X"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "1fe84cdc92a5136ff8da7cb6150b2dda5723bc41d62dd429df6d15ae1d54bf80"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "本輪用Bernoulli參數p_n=p+c/(n+s)的symbolic tail並限制於0到1。"
    },
    {
      "id": "a2",
      "zh-Hant": "只核極限CDF連續點與明示threshold。"
    }
  ],
  "symbols": [
    {
      "token": "D",
      "role": "convergence_in_distribution",
      "role_zh": "依分布收斂",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "cdf",
      "role_zh": "CDF",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-cumulative-distribution-function",
      "title_zh": "累積分布函數",
      "reason_zh": "以CDF連續點表述。"
    },
    {
      "id": "mko-function-limit",
      "title_zh": "函數極限",
      "reason_zh": "逐點極限責任。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "F_n按n索引。"
    },
    {
      "id": "mko-finite-joint-distribution",
      "title_zh": "有限聯合分布",
      "reason_zh": "離散law提供CDF。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "只在limit CDF連續點要求。",
      "共同coupling不是定義必要條件。",
      "symbolic parameter tail提供全稱證書。"
    ],
    "common_misconception_zh": "核有限n、要求jump點也收斂，或直接推依機率收斂。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理明示Bernoulli symbolic family；不推更強收斂模式。",
    "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Portmanteau與CLT。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-convergence-in-distribution.md 整理；2026-10-14 脈衝 R03（使用者手動接續 2026-10-31 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"
  }
}