convergence in probability iff expectation...

**Beaky** · December 12th 2010, 09:26 PM

Suppose $Y_{n}\ge 0$ . Show $Y_{n}\rightarrow 0$ in probability iff $E(\frac{Y_{n}}{1+Y_{n}})\rightarrow 0$ .

Any tips would be much appreciated.

**Focus** · December 13th 2010, 05:23 AM

What have you done so far? Some basic hints, Markov's inequality will be useful and also splitting the expectation.

**Beaky** · December 13th 2010, 10:34 AM

I was trying with Markov's inequality earlier but didn't get anywhere. I think I've managed to show one inclusion now. I don't see how to split the expectation either which is maybe why I'm stuck on the other.

$P(\frac{Y_{n}}{1+Y_{n}}\ge\epsilon)=P(Y_{n}\ge\eps ilon+\epsilon Y_{n})=P(Y_{n}\ge\frac{\epsilon}{1-\epsilon})\le E(\frac{Y_{n}}{1+Y_{n}})/\epsilon$

So I have expectation >> convergence since $\frac{\epsilon}{1-\epsilon}<\epsilon$ for small $\epsilon$ .

**Focus** · December 13th 2010, 02:43 PM

Originally Posted by Beaky

I was trying with Markov's inequality earlier but didn't get anywhere. I think I've managed to show one inclusion now. I don't see how to split the expectation either which is maybe why I'm stuck on the other.

$P(\frac{Y_{n}}{1+Y_{n}}\ge\epsilon)=P(Y_{n}\ge\eps ilon+\epsilon Y_{n})=P(Y_{n}\ge\frac{\epsilon}{1-\epsilon})\le E(\frac{Y_{n}}{1+Y_{n}})/\epsilon$

So I have expectation >> convergence since $\frac{\epsilon}{1-\epsilon}<\epsilon$ for small $\epsilon$ .

You are supposed to show that $Y_n \rightarrow 0$ which is pretty similar to what you did if you notice that $\frac{Y_n}{Y_n+1} \leq Y_n$ .

For the converse try splitting as follows
$\mathbb{E}[|Y_n/(Y_n+1)|]\leq \mathbb{E}[|Y_n/(Y_n+1)|\mathbf{1}_{|Y_n|>\epsilon}]+\mathbb{E}[|Y_n/(Y_n+1)|\mathbf{1}_{|Y_n|\leq \epsilon}]$

You can bound the RHS term by epsilon (as I indicated above), the second term you can split further by using the fact that $Y_n/(Y_n+1) =1-1/(Y_n+1) \leq 1$ then use convergence in probability.

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