第 15398 題：有限Martingale

原題與已知條件

同一有限probability space filtration與process。操作=martingale-qualification；資料={"masses":[[1,4],[1,4],[1,4],[1,4]],"partitions":[[0,0,0,0],[0,0,1,1]],"process_values":[[10,10,10,10],[10,10,10,10]]}。請依 [nesting_failures,adaptedness_failures,conditional_steps,martingale] 欄序回答。

• G001 — 同一有限probability space filtration與process。操作=martingale-qualification；資料={"masses":[[1,4],[1,4],[1,4],[1,4]],"partitions":[[0,0,0,0],[0,0,1,1]],"process_values":[[10,10,10,10],[10,10,10,10]]}。請依 [nesting_failures,adaptedness_failures,conditional_steps,martingale] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-martingale

逐步推導

martingale full probability mass := sum(1/4, 1/4, 1/4, 1/4)

martingale full probability mass = 1

same finite probability carrier size=4;exact masses=[1/4, 1/4, 1/4, 1/4];partitions in time order=[[0, 0, 0, 0], [0, 0, 1, 1]]

total mass=1;all later-equal pairs with earlier-different labels=[]

time=0;block label=0;indices=[0, 1, 2, 3];current values in original order=[10, 10, 10, 10]

distinct current values first-encounter order=[10];constant on block=是

time=1;block label=0;indices=[0, 1];current values in original order=[10, 10]

distinct current values first-encounter order=[10];constant on block=是

time=1;block label=1;indices=[2, 3];current values in original order=[10, 10]

distinct current values first-encounter order=[10];constant on block=是

martingale time0 block0 mass := sum(1/4, 1/4, 1/4, 1/4)

martingale time0 block0 mass = 1

martingale time0 block0 next weighted outcome0 := mul(1/4, 10)

martingale time0 block0 next weighted outcome0 = 5/2

martingale time0 block0 next weighted outcome1 := mul(1/4, 10)

martingale time0 block0 next weighted outcome1 = 5/2

martingale time0 block0 next weighted outcome2 := mul(1/4, 10)

martingale time0 block0 next weighted outcome2 = 5/2

martingale time0 block0 next weighted outcome3 := mul(1/4, 10)

martingale time0 block0 next weighted outcome3 = 5/2

martingale time0 block0 next weighted sum := sum(5/2, 5/2, 5/2, 5/2)

martingale time0 block0 next weighted sum = 10

martingale time0 block0 next conditional mean := div(10, 1)

martingale time0 block0 next conditional mean = 10

time=0;block label=0;current distinct values=[10];next weighted sum=10;positive block mass=1

conditional mean=10;conditional step row=[0, 0, 10, [10], 是]

nesting failures=[];adaptedness failures=[];conditional rows=[[0, 0, 10, [10], 是]]

finite integrable process;all three qualifications=是

逐步反向核對

• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next conditional mean，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted sum，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome3，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome2，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome1，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 next weighted outcome0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 martingale time0 block0 mass，不以小數近似替代。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 以精確有理數重算 martingale full probability mass，不以小數近似替代。

結論

([], [], [[0, 0, 10, [10], 是]], 是)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。