# Thread: Strong Law of Large Numbers Application

1. ## Strong Law of Large Numbers Application

This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

Problem:
Deﬁne the sequence $X_{n}$ inductively by setting $X_{0} = 1$, and selecting $X_{n+1}$ randomly and uniformly from the interval $[0, X_{n}]$. Prove that $n^{-1}$ $\log$ $X_{n}$ converges almost surely to a constant, and evaluate the limit.

Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):
let $\log$ $X_{n}=$ $\log$ $X_{n} -$ $\log$ $X_{n-1} +$ $\log$ $X_{n-1} + ... +$ $\log$ $X_{1} +$ $\log$ $X_{0}$

2. ## Re: Strong Law of Large Numbers Application

I don't see a SLLN here.
Why don't you obtain the distribution of $X_n$ and then take the log of that rv?

3. ## Re: Strong Law of Large Numbers Application

Originally Posted by puggles
This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

Problem:
Deﬁne the sequence $X_{n}$ inductively by setting $X_{0} = 1$, and selecting $X_{n+1}$ randomly and uniformly from the interval $[0, X_{n}]$. Prove that $n^{-1}$ $\log$ $X_{n}$ converges almost surely to a constant, and evaluate the limit.

Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):
let $\log$ $X_{n}=$ $\log$ $X_{n} -$ $\log$ $X_{n-1} +$ $\log$ $X_{n-1} + ... +$ $\log$ $X_{1} +$ $\log$ $X_{0}$
With $\ln X_{n}$ You mean $E \{\ln X_{n}\}$... don't You?...

Kind regards

$\chi$ $\sigma$

4. ## Re: Strong Law of Large Numbers Application

Originally Posted by chisigma
With $\ln X_{n}$ You mean $E \{\ln X_{n}\}$... don't You?...
If the answer is 'yes', then the quantity $\mu_{n}= E \{X_{n}\}$ is the solution of the difference equation...

$\mu_{n+1}= \frac{\mu_{n}}{2}\ ,\ \mu_{0}=1$ (1)

... so that is...

$\mu_{n}= \frac{1}{2^{n}}$ (2)

Kind regards

$\chi$ $\sigma$

5. ## Re: Strong Law of Large Numbers Application

Originally Posted by chisigma
If the answer is 'yes', then the quantity $\mu_{n}= E \{X_{n}\}$ is the solution of the difference equation...

$\mu_{n+1}= \frac{\mu_{n}}{2}\ ,\ \mu_{0}=1$ (1)

... so that is...

$\mu_{n}= \frac{1}{2^{n}}$ (2)
... and setting $\lambda_{n}= E \{\ln X_{n}\}$ is...

$\lambda_{n+1}= \frac{1}{X_{n}}\ \int_{0}^{X_{n}} \ln x\ dx = \lambda_{n}-1\ ,\ \lambda_{0}=0$ (1)

... so that is...

$\lambda_{n}= -n$ (2)

Kind regards

$\chi$ $\sigma$

6. ## Re: Strong Law of Large Numbers Application

Thanks everyone for your help. Sorry I took so long to respond... the assignment is already turned in, but your comments have helped me understand this much better! Thank you