摘要
In a simple random walk {S_n}_n≥0, let L_n = min_(0≤i≤n) S_i and R_n = max_(0≤i≤n) S_i. Suppose {a_n}_n≥1 and {b_n}_n≥1 are strictly increasing positive integer sequences and let K = {[-a_n; b_n] | n ∈ N}. We are interested in the probability that there is an n such that [L_n;R_n] ∈ K. If {S_n}_(n≥0) is symmetric, it is derived that P([L_n;R_n] ∈ K for some n) = (The equation is abbreviated) which implies that P([L_n;R_n] ∈ K for some n) = 1 if (The equation is abbreviated). If {S_n}_(n≥0) is asymmetric and K_0 = {[-n; n] |n ∈ N} we can also compute the probability that there is an n such that [L_n;R_n] ∈ K_0.
並列摘要
在簡單隨機漫步{S_n}_n≥0中, 令Ln = min_(0≤i≤n)S_i以及R_n = max_(0≤i≤n)S_i。假設{a_n}_n≥1和{b_n}_n≥1是兩個嚴格遞增的正整數數列,並且令K = {[-a_n; b_n]|n ∈ N}。我們感興趣的事是[L_n;R_n]在某些時間n屬於K的機率。如果{S_n}_(n≥0)是對稱的,可求得P([L_n;R_n] ∈ K for some n) =(方程式略)，這導致如果(方程式略)，則P([L_n;R_n] ∈K for some n) = 1。如果{S_n}_(n≥0)是不對稱的, 那麼令K_0 = {[-n; n]|n ∈N}, 我們也可以計算出[L_n;R_n]在某些時間n屬於K_0的機率。