ABSTRACT

Another way of modelling the gambler’s ruin problem of the last chapter is as a one-dimensional random walk. Suppose that a + 1 $ a+1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_1.tif"/> positions are marked out on a straight line and numbered 0 , 1 , 2 , … , a $ 0,1,2,\ldots ,a $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_2.tif"/> . A person starts at k where 0 < k < a $ 0< k< a $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_3.tif"/> . The walk proceeds in such a way that at each step there is a probability p that the walker goes ‘forward’ one place to k + 1 $ k+1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_4.tif"/> , and a probability q = 1 - p $ q=1-p $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_5.tif"/> that the walker goes ‘back’ one place to k - 1 $ k-1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_6.tif"/> . The walk continues until either 0 or a is reached, and then ends. Generally, in a random walk, the position of a walker after having moved n times is known as the state of the walk after n steps or after covering n stages. Thus the walk described above starts at stage k at step 0 and moves to either stage k - 1 $ k-1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_7.tif"/> or stage k + 1 $ k+1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_8.tif"/> after 1 step, and so on. A random walk is said to be symmetric if p = q = 1 2 $ p=q=\frac{1}{2} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315156576/49190045-e11a-4825-88bf-d327626f56ef/content/inline-math3_9.tif"/> .