Chapter 4: E4.10 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Star Trek, 2

See E10.11, which you can do now.

E10.11 $^{*}$ . Star Trek, 2. ’Captain’s Log...

Mr Spock and Chief Engineer Scott have modified the control system so that the Enterprise is confined to move for ever in a fixed plane passing through the Sun. However, the next ’hop-length’ is now automatically set to be the current distance to the Sun (’next’ and ’current’ being updated in the obvious way). Spock is muttering something about logarithms and random walks, but I wonder whether it is (almost) certain that we will get into the Solar System sometime...’

Hint. Let $X_{n}:=\log R_{n}-\log R_{n-1}$ . Prove that $X_{1},X_{2},\dotsc$ is an lID sequence of variables each of mean $0$ and finite variance $\sigma^{2}$ (say), where $\sigma>0$ . let $S_{n}:=X_{1}+X_{2}+\dotsb+X_{n}.$ Prove that if $\alpha$ is a fixed positive number, then $\begin{align*} \mathbb{P}[\inf_{n}S_{n}=-\infty] & \ge\mathbb{P}[S_{n}\le-\alpha\sigma\sqrt{n},\text{ i.o.}]\\ \ge\limsup\mathbb{P}[S_{n} & \le-\alpha\sigma\sqrt{n}J=\Phi(-\alpha)>O. \end{align*}$ (Use the Central Limit Theorem.) Prove that the event $\{\inf_{n}S_{n}=-\infty\}$ is in the tail $\sigma$ -algebra of the $(X_{n})$ sequence.

If the Sun is at the origin, and the spaceship is at $(R_{n},0)$ , then the new position of the spaceship is at $R_{n}(1+\cos\phi,\sin\phi)$ where $\phi\sim U[0,2\pi]$ , and the square of the new distance is $R_{n+1}^{2}=4R_{n}^{2}\cos^{2}\tfrac{\phi}{2}$ , so that $\begin{align*} \log R_{n+1}-\log R_{n} & =\log2+\log|\cos\tfrac{\phi}{2}|\\ & =\sum_{k\in\mathbb{N}}(-1)^{k-1}\tfrac{\cos(k\phi)}{k}. \end{align*}$ Hence, we get a series of IID random variables of $0$ expectation and finite variance.

Now, if there is a sequence $\{n_{k}\}$ such that for $k\in\mathbb{N}$ , $\mathbb{P}(S_{n_{k}}\le-\alpha\sigma\sqrt{n_{k}})\ge p,$ then $\mathbb{P}(S_{n}>-\alpha\sigma\sqrt{n},\text{ ev})\le1-p,$ hence, $\begin{align*} \mathbb{P}(\inf_{n}S_{n}=-\infty) & \ge\mathbb{P}(S_{n}\le-\alpha\sigma\sqrt{n},\text{ i.o.})\\ & \ge\limsup\mathbb{P}(S_{n}\le-\alpha\sigma\sqrt{n})\\ \text{(by the CLT)} & =\Phi(-\alpha)\\ & >0. \end{align*}$ Further, for a fixed $k\in\mathbb{N}$ , the event $\{\inf_{n}S_{n}=-\infty\}$ equals $\{\inf_{n\ge k}(S_{n}-S_{k})=-\infty\}$ , hence, this event is in the tail $\sigma$ -algebra of the sequence $\{X_{n}\}$ , implying that the spaceship will a.s. come to the Solar system.

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Chapter 4: E4.10 Solution

Chapter 4: E4.10 Solution