本篇論文主要的目的是希望能夠以賭徒破產理論為基礎，三維空間中的轉 軸公式和截面圓性質為輔助，建立一個簡單的演算法，模擬出三維空間中在球 表面上的隨機漫步過程，並藉此進一步了解隨機漫步在沒有限制方向但步長固 定的情況下，在球的表面上會以什麼樣的情況進行。由本篇論文提供的演算 法，我們可以得到隨機漫步過程各點在經、緯度或直角座標移動的情況，同時 能夠計算出成功抵達目的地的機率和整個漫步過程所花費的時間，即：期望步 數。
由模擬出的數值我們發現，在固定步長以及下一步方向服從特定分配的球 面上隨機漫步與賭徒破產定理公式具有相似的趨勢，即：當向上向下的機率比 值等於1時，成功抵達目的地(北極圈)的機率與起始位置的緯度成正比關係，
且整個隨機漫步所花費的時間與起始位置的緯度成二次曲線的關係，其中以位 於目標緯度1/2的緯度為起始位置者所需花費的時間最多。而當向上向下的機 率比值不等於1時，成功抵達目的地的機率和整個隨機漫步所花費時間的結果 也與賭徒破產公式的結果有相同的趨勢。但球面上的隨機漫步因為每一步向上 或向下移動的垂直距離都不同，因此花費的時間都更長，而且具有因橫跨緯度 範圍不同而產生隨起始緯度遞增，與賭徒破產理論值的差異越大(當p<0.5)或越 小(當p>0.5)的性質。
由於本篇論文中所建立隨機漫步方式，具有緯度越高向北所能夠移動的最 長距離越短的特性，意即：緯度越高越不容易向上爬升；近似於股票市場中，
當指數越接近歷史高點，漲幅越小的情況，因此希望所提出的模擬方法，未來 能夠在相關領域加以應用。
參考文獻
［1］ Arfken, G., Mathematical Methods for Physicists, 3rd edition, Academic Press, Orlando, FL, 1985.
［2］ Coxeter, H.S.M., Introduction to Geometry, 2nd ed, Wiley, New York, 1969.
［3］ Dyutiman Das, "Quantum Monte Carlo With a Stochastic Potential solver", University of Illinois, degree of Doctor, 2005
［4］ Evans, M.; Hastings, N.; and Peacock, B., Statistical Distributions, 3rd edition, Wiley, New York, 2000
［5］ J.A. Given, J.B. Hubbard, J.F. Douglas, "A first-passage algorithm for the hydrodynamic friction and diffusion-limited reaction rate of
macromolecules", J.Chem. Phys. 106, , 3721–3771, 1997.
［6］ Mascagni M. and C.-O. Hwang. "ε-shell error analysis in “Walk On Spheres” algorithms", Math. Comput. Simulation, in press, 2002.
［7］ Muller M. E., "Some Continuous Monte Carlo Methods for the Dirichlet Problem", Annals of Mathematical Statistics, 27, 567-589 ,1956.
［8］ Roberts, P.H. and Ursell, H.D., "Random walk on a sphere and on a Riemannian manifold", PhUos. Trans. A ,Mathematical .and Physical Sciences ,252, 317-356, 1960.
［9］ Sheldon M. Ross, Stochastic Process, 2nd edition, WILEY, 1980.
［10］ Thisted, R. A., Elements of Statistical Computing, Chapman and Hall, New York, 1988.
附 錄 一、
表3.1 緯度對映成功抵達北極圈機率模擬結果
緯度 成功機率 緯度 成功機率 緯度 成功機率
1 0.0144 23 0.2626 45 0.5537 2 0.0247 24 0.2756 46 0.5833 3 0.0391 25 0.2845 47 0.5894 4 0.0517 26 0.3018 48 0.6012 5 0.0621 27 0.3146 49 0.6143 6 0.072 28 0.3292 50 0.6364 7 0.0801 29 0.3315 51 0.6574 8 0.0932 30 0.3504 52 0.6824 9 0.0998 31 0.3702 53 0.6972 10 0.1126 32 0.3836 54 0.7138 11 0.124 33 0.3892 55 0.7344 12 0.1307 34 0.3988 56 0.7547 13 0.1497 35 0.4195 57 0.779 14 0.1623 36 0.4247 58 0.7928 15 0.1784 37 0.4321 59 0.8177 16 0.1799 38 0.4482 60 0.8286 17 0.1884 39 0.4725 61 0.8537 18 0.2088 40 0.4852 62 0.8826 19 0.2158 41 0.5034 63 0.9091 20 0.2328 42 0.5223 64 0.9252 21 0.2388 43 0.5376 65 0.9561 22 0.2431 44 0.545
表3.2 緯度對期望步數模擬結果
緯度 期望步數 緯度 期望步數 緯度 期望步數
1 185.7869 23 2132.031 45 2248.373 2 317.0791 24 2143.494 46 2187.003 3 430.5564 25 2202.633 47 2178.715 4 552.4942 26 2244.336 48 2093.866 5 670.5594 27 2280.125 49 2084.278 6 798.1686 28 2347.84 50 2017.356 7 902.2981 29 2387.121 51 1927.679 8 1012.038 30 2381.418 52 1847.946 9 1071.853 31 2391.437 53 1759.81 10 1199.587 32 2412.104 54 1711.938 11 1270.092 33 2454.929 55 1609.169 12 1355.005 34 2450.352 56 1502.505 13 1486.08 35 2449.169 57 1386.755 14 1568.415 36 2459.602 58 1298.695 15 1623.881 37 2426.542 59 1167.023 16 1664.632 38 2417.266 60 1081.353 17 1787.097 39 2432.37 61 940.4222 18 1850.396 40 2403.899 62 768.4553 19 1905.596 41 2358.658 63 610.9491 20 2002.244 42 2332.806 64 513.36 21 2023.779 43 2314.562 65 310.1307 22 2087.072 44 2273.538
表3.3 緯度對成功機率值的理論值與模擬結果比較
表3.4 緯度對停止步數的期望值理論值與模擬結果比較 1 91.78571 91.69662 0.000971 34 80 79.9395 0.000756 2 129.6939 129.7817 0.000677 35 77.5 77.50962 0.000124 3 144.5117 144.311 0.001388 36 75 74.97604 0.000319 4 149.4336 149.3787 0.000367 37 72.5 72.56172 0.000851 5 150.1144 150.0829 0.000209 38 70 70.05252 0.00075 6 148.9776 149.238 0.001748 39 67.5 67.574 0.001096 7 147.0618 147.0515 6.99E-05 40 65 65.01852 0.000285 8 144.8122 144.8428 0.000211 41 62.5 62.48858 0.000183 9 142.4195 142.3525 0.000471 42 60 59.95134 0.000811 10 139.9655 139.9507 0.000106 43 57.5 57.4767 0.000405 11 137.4852 137.4368 0.000352 44 55 54.98498 0.000273 12 134.9937 135.1055 0.000828 45 52.5 52.49672 6.25E-05 13 132.4973 132.5052 5.96E-05 46 50 49.96878 0.000624 14 129.9988 129.954 0.000345 47 47.5 47.56354 0.001338 15 127.4995 127.4178 0.000641 48 45 45.0165 0.000367 16 124.9998 125.0077 6.3E-05 49 42.5 42.5962 0.002264 17 122.4999 122.4856 0.000117 50 40 40.0061 0.000153 18 120 119.8916 0.000903 51 37.5 37.48846 0.000308 19 117.5 117.6291 0.001099 52 35 35.04942 0.001412 20 115 114.9007 0.000863 53 32.5 32.47774 0.000685 21 112.5 112.578 0.000693 54 30 30.00688 0.000229 22 110 110.0418 0.00038 55 27.5 27.46874 0.001137 23 107.5 107.4801 0.000185 56 25 24.9664 0.001344 24 105 105.0422 0.000402 57 22.5 22.48946 0.000468 25 102.5 102.4697 0.000296 58 20 20.01906 0.000953 26 100 100.1001 0.001001 59 17.5 17.52362 0.00135 27 97.5 97.60414 0.001068 60 15 15.04154 0.002769 28 95 94.9919 8.53E-05 61 12.5 12.4802 0.001584 29 92.5 92.40544 0.001022 62 10 10.01566 0.001566 30 90 90.0864 0.00096 63 7.5 7.48448 0.002069 31 87.5 87.55426 0.00062 64 5 5.015 0.003 32 85 84.91768 0.000968 65 2.5 2.38083 0.047668
表3.6 不同beta分配緯度對成功機率模擬結果
緯度 Beta(3,1) Beta(3,2) Beta(3,3) Beta(2,3) Beta(1,3) uniform
1 0 0 0.0155 0.8081 0.9786 0.0144 2 0 0 0.0262 0.9374 0.9981 0.0247 3 0 0 0.0361 0.9792 0.9998 0.0391
4 0 0 0.0469 0.9927 1 0.0517
5 0 0 0.0594 0.9977 1 0.0621
6 0 0 0.0706 0.9991 1 0.072
7 0 0 0.0849 0.9999 1 0.0801
8 0 0 0.0897 1 1 0.0932
9 0 0 0.1055 1 1 0.0998
10 0 0 0.119 1 1 0.1126
11 0 0 0.1312 1 1 0.124
12 0 0 0.1385 1 1 0.1307
13 0 0 0.1543 1 1 0.1497
14 0 0 0.1601 1 1 0.1623
15 0 0 0.1789 1 1 0.1784
16 0 0 0.1891 1 1 0.1799
17 0 0 0.1968 1 1 0.1884
18 0 0 0.2084 1 1 0.2088
19 0 0 0.2217 1 1 0.2158
20 0 0 0.2271 1 1 0.2328
21 0 0 0.253 1 1 0.2388
22 0 0 0.2613 1 1 0.2431
23 0 0 0.2614 1 1 0.2626
24 0 0 0.283 1 1 0.2756
25 0 0 0.2833 1 1 0.2845
26 0 0 0.3082 1 1 0.3018
27 0 0 0.3194 1 1 0.3146
28 0 0 0.325 1 1 0.3292
29 0 0 0.3518 1 1 0.3315
30 0 0 0.3571 1 1 0.3504
31 0 0 0.3703 1 1 0.3702
32 0 0 0.3888 1 1 0.3836
33 0 0 0.3945 1 1 0.3892
34 0 0 0.4088 1 1 0.3988
35 0 0 0.4183 1 1 0.4195
36 0 0 0.4327 1 1 0.4247
37 0 0 0.4523 1 1 0.4321
38 0 0 0.4577 1 1 0.4482
39 0 0 0.4813 1 1 0.4725
40 0 0 0.4987 1 1 0.4852
41 0 0 0.5113 1 1 0.5034
42 0 0 0.524 1 1 0.5223
43 0 0 0.5355 1 1 0.5376
44 0 0 0.5578 1 1 0.545
45 0 0 0.5686 1 1 0.5537
46 0 0 0.5888 1 1 0.5833
47 0 0 0.5967 1 1 0.5894
48 0 0 0.618 1 1 0.6012
49 0 0 0.6341 1 1 0.6143
50 0 0 0.6537 1 1 0.6364
51 0 0 0.6655 1 1 0.6574
52 0 0 0.6784 1 1 0.6824
53 0 0 0.7045 1 1 0.6972
54 0 0 0.7142 1 1 0.7138
55 0 0 0.7323 1 1 0.7344
56 0 0 0.763 1 1 0.7547
57 0 0 0.7761 1 1 0.779
58 0 0 0.8001 1 1 0.7928
59 0 0 0.8164 1 1 0.8177
60 0 0 0.8479 1 1 0.8286
61 0 0.0007 0.864 1 1 0.8537
62 0.0001 0.003 0.8871 1 1 0.8826 63 0.0002 0.0125 0.9052 1 1 0.9091 64 0.0008 0.0397 0.9322 1 1 0.9252 65 0.0105 0.13 0.9574 1 1 0.9561
表3.7 不同beta分配緯度對停止步數的期望值模擬結果 緯度 Beta(3,1) Beta(3,2) Beta(3,3) Beta(2,3) Beta(1,3) uniform
1 3.0107 5.0035 180.3125 184.3015 135.593 185.7869 2 5.0573 8.2842 319.8471 211.2709 136.3318 317.0791 3 7.0851 11.5375 411.3669 217.0764 134.4115 430.5564 4 9.1766 15.2498 526.7513 216.8748 132.3252 552.4942 5 11.2345 18.4776 637.5655 214.7779 130.2504 670.5594 6 13.3317 21.8515 759.7591 211.4438 128.1467 798.1686 7 15.4696 25.5045 886.9566 208.3564 126.3201 902.2981 8 17.5481 28.7266 972.8154 204.6057 124.0601 1012.038 9 19.6669 32.1537 1060.774 201.2125 122.0637 1071.853 10 21.6731 35.6823 1172.046 197.6118 119.795 1199.587 11 23.8364 39.1877 1280.919 195.088 117.8662 1270.092 12 25.8858 42.4798 1363.118 190.2699 115.6743 1355.005 13 27.9707 46.0516 1439.68 187.5566 113.6164 1486.08 14 30.0505 49.0233 1483.542 183.2318 111.5113 1568.415 15 32.1885 52.8233 1603.091 180.1565 109.1784 1623.881 16 34.23 56.3425 1641.862 177.3172 107.2223 1664.632 17 36.2197 59.7091 1707.947 173.5929 105.1063 1787.097 18 38.3903 63.2145 1794.412 170.2867 102.9212 1850.396 19 40.6409 66.6959 1860.014 166.7295 100.7224 1905.596 20 42.4517 70.0618 1873.479 163.1231 98.7095 2002.244 21 44.7391 73.1979 1967.313 159.7736 96.5861 2023.779 22 46.8565 76.5753 2014.199 156.3728 94.5989 2087.072 23 48.7689 79.9759 2061.144 152.7598 92.4471 2132.031 24 50.9784 83.8929 2123.897 149.3525 90.4005 2143.494 25 53.1118 86.8056 2157.574 146.1692 88.3166 2202.633 26 55.0976 90.5261 2207.049 142.6508 86.3128 2244.336 27 57.238 93.8755 2234.421 138.785 84.1359 2280.125 28 59.4122 96.9957 2216.109 136.1427 82.0358 2347.84 29 61.4157 101.1855 2301.971 131.8018 79.8468 2387.121 30 63.4214 103.8144 2315.958 128.2073 77.6822 2381.418 31 65.6423 107.9886 2332.606 125.0758 75.8923 2391.437 32 67.6498 111.2967 2331.292 121.6061 73.5955 2412.104 33 69.827 114.4198 2366.955 118.4889 71.3944 2454.929 34 71.7706 117.7441 2333.543 114.7886 69.4419 2450.352 35 73.7005 121.2781 2396.634 110.9406 67.1696 2449.169 36 76.1081 124.7699 2394.805 107.725 65.1306 2459.602
37 78.1078 128.3355 2358.249 104.4098 63.0472 2426.542 38 80.0623 131.4538 2351.234 100.6605 60.9171 2417.266 39 82.1415 135.3498 2306.715 97.5422 58.6251 2432.37 40 84.2802 138.3091 2288.097 93.6528 56.7282 2403.899 41 86.4155 141.8152 2317.528 90.6306 54.5682 2358.658 42 88.4069 145.0424 2257.72 87.0748 52.5097 2332.806 43 90.3704 148.3557 2267.53 83.3361 50.331 2314.562 44 92.6086 151.919 2218.134 80.2903 48.3238 2273.538 45 94.5109 155.7997 2172.059 76.2367 46.0043 2248.373 46 96.8082 158.6885 2133.831 72.7141 44.1103 2187.003 47 98.9383 162.0521 2117.097 69.2155 41.8842 2178.715 48 100.6985 165.6327 2023.482 65.8812 39.7505 2093.866 49 102.9291 169.1571 1978.133 62.3041 37.7662 2084.278 50 104.9258 172.1504 1920.154 58.929 35.5115 2017.356 51 107.0889 175.2372 1846.762 55.2657 33.5173 1927.679 52 109.1558 179.0327 1820.404 51.9124 31.2601 1847.946 53 111.2604 182.352 1704.808 48.2845 29.1893 1759.81 54 113.4775 185.9548 1652.615 45.0079 27.053 1711.938 55 115.2988 189.3317 1546.27 41.3603 24.9659 1609.169 56 117.5313 192.9857 1468.109 37.8668 22.7669 1502.505 57 119.5189 195.6791 1368.416 34.0731 20.6828 1386.755 58 121.6093 198.8116 1219.535 31.0518 18.5684 1298.695 59 123.4326 203.2406 1151.206 27.3145 16.4353 1167.023 60 125.7872 205.8612 1019.12 23.945 14.2769 1081.353 61 127.8702 209.4816 888.3129 20.1534 12.1765 940.4222 62 129.8617 211.337 738.9128 16.608 10.1191 768.4553 63 131.8182 213.5076 620.4948 13.1995 7.9119 610.9491 64 133.7293 210.7151 446.6872 9.603 5.7467 513.36 65 134.5276 193.0658 294.8881 6.1457 3.6641 310.1307