{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-discrete-stochastic-process",
  "semantic_family_id": "sf-discrete-stochastic-process",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "離散隨機過程"
  },
  "summary": {
    "zh-Hant": "離散隨機過程是共同機率空間上的離散時間隨機變數族；邊際分布不決定聯合律。"
  },
  "statement": {
    "zh-Hant": "離散時間隨機過程(X_n)是同一機率空間上的隨機變數族。本輪以有限時域完整路徑分布指定joint law，從而導出各時刻marginal與兩時刻joint；這不自動給Markov性或stationarity。"
  },
  "formula": {
    "tex": "X(n)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>X</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "X",
        "glyph": "X"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "n",
          "glyph": "n"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0f02048423626a13b2324a0a0e227699333e642cd75f8e24c498f7352512c1b5"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫列出互異、等長的完整有限路徑support與精確PMF。"
    },
    {
      "id": "a2",
      "zh-Hant": "有限時域joint law不外推無限時間、轉移律、stationarity或ergodicity。"
    }
  ],
  "symbols": [
    {
      "token": "X",
      "role": "stochastic_process",
      "role_zh": "離散隨機過程",
      "scope": "representative_formula"
    },
    {
      "token": "n",
      "role": "discrete_time",
      "role_zh": "離散時間索引",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-discrete-probability-distribution",
      "title_zh": "有限離散機率分布",
      "reason_zh": "完整路徑support具有精確PMF。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "時間以自然數離散索引。"
    },
    {
      "id": "mko-random-variable",
      "title_zh": "隨機變數",
      "reason_zh": "每個時刻變數位於共同機率空間。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "完整路徑law同時指定時間間依賴。",
      "marginal由固定時刻聚合路徑質量取得，但不同joint law可有相同marginals。",
      "Markov性需條件分布責任，不能從marginal表推得。"
    ],
    "common_misconception_zh": "把每時刻邊際分布集合當完整隨機過程，或自動宣稱Markov。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限時域完整joint path 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": "加入有限Markov鏈、轉移矩陣與路徑機率。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-discrete-stochastic-process.md 整理；2026-10-13 脈衝 R06（使用者手動接續 2026-10-23 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}