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Omni-MATH / Let {X_(n)}_(n≥1) be i.i.d. random variables such that P(X₁=1)=1−P(X₁=−1)=p>(1)/(2). Let S₀=0,S_(n)=∑_(i=1)^(n)X_(i). Define the range of {S_(n)}_(n≥0) by R_(n)=#{S₀,S₁,S₂,⋯,S_(n)}…

Problem

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problem

Let {Xn}n≥1\left\{X_{n}\right\}_{n \geq 1} be i.i.d. random variables such that P(X1=1)=1−P(X1=−1)=p>12\mathbb{P}\left(X_{1}=1\right)=1-\mathbb{P}\left(X_{1}=-1\right)=p>\frac{1}{2}. Let S0=0,Sn=∑i=1nXiS_{0}=0, S_{n}=\sum_{i=1}^{n} X_{i}. Define the range of {Sn}n≥0\left\{S_{n}\right\}_{n \geq 0} by Rn=#{S0,S1,S2,⋯,Sn}R_{n}=\#\left\{S_{0}, S_{1}, S_{2}, \cdots, S_{n}\right\}, which is the number of distinct points visited by the random walk {Sn}n≥0\left\{S_{n}\right\}_{n \geq 0} up to time nn. (1) Prove E(Rn)=E(Rn−1)+P(S1S2⋯Sn≠0),n=1,2,⋯\mathbb{E}\left(R_{n}\right)=\mathbb{E}\left(R_{n-1}\right)+P\left(S_{1} S_{2} \cdots S_{n} \neq 0\right), \quad n=1,2, \cdots. (2) Find limn→∞1nE(Rn)\lim _{n \rightarrow \infty} \frac{1}{n} \mathbb{E}\left(R_{n}\right).
Plain-text mathematical notation (without MathML)
Let {X_(n)}_(n≥1) be i.i.d. random variables such that P(X₁=1)=1−P(X₁=−1)=p>(1)/(2). Let S₀=0,S_(n)=∑_(i=1)^(n)X_(i). Define the range of {S_(n)}_(n≥0) by R_(n)=#{S₀,S₁,S₂,⋯,S_(n)}, which is the number of distinct points visited by the random walk {S_(n)}_(n≥0) up to time n. (1) Prove E(R_(n))=E(R_(n−1))+P(S₁S₂⋯S_(n)≠0), n=1,2,⋯. (2) Find lim_(n→∞)(1)/(n)E(R_(n)).
Original LaTeX notation
Let \(\left\{X_{n}\right\}_{n \geq 1}\) be i.i.d. random variables such that \(\mathbb{P}\left(X_{1}=1\right)=1-\mathbb{P}\left(X_{1}=-1\right)=p>\frac{1}{2}\). Let \(S_{0}=0, S_{n}=\sum_{i=1}^{n} X_{i}\). Define the range of \(\left\{S_{n}\right\}_{n \geq 0}\) by \(R_{n}=\#\left\{S_{0}, S_{1}, S_{2}, \cdots, S_{n}\right\}\), which is the number of distinct points visited by the random walk \(\left\{S_{n}\right\}_{n \geq 0}\) up to time \(n\). (1) Prove \(\mathbb{E}\left(R_{n}\right)=\mathbb{E}\left(R_{n-1}\right)+P\left(S_{1} S_{2} \cdots S_{n} \neq 0\right), \quad n=1,2, \cdots\). (2) Find \(\lim _{n \rightarrow \infty} \frac{1}{n} \mathbb{E}\left(R_{n}\right)\).

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Source and history

task_id
800f6e35-3b97-5749-94cc-f9c495055885
task_key
test--800f6e35-3b97-5749-94cc-f9c495055885
task_revision_id
1
config / split
/ test

Official source

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