Understanding the Proof: Why Does It Work?

[GabrielN00]

Homework Statement

2. Questions

I want to know how this proof works, but there are several things I don't understand.

(1) How does the $k\epsilon$ appears? $k$ is the lipschitz constant, but what about $\epsilon$? Is the author taking $||x-y||<\epsilon$?

(2) What does the subscript $1_\{||X_n-Y_n||>\epsilon\}$ means and why is it there? I think the author divided the those values who were at a distance smaller than $\epsilon$ and those who weren't, but I'm not completely sure about this.

(3) Why is it $E\{|f(X_n)-f(Y)n|1_{||X_n-Y_n||>\epsilon}\}\leq 2\alpha P\{||X_n-Y_n||\}$? This might be related to my second question, I am not completely sure about the meaning of the subscript. I see $f$ is bounded by $\alpha$, but why is it $2\alpha$?

 

[Mark44]

Homework Statement

View attachment 213380

2. Questions

I want to know how this proof works, but there are several things I don't understand.

(1) How does the $k\epsilon$ appears? $k$ is the lipschitz constant, but what about $\epsilon$? Is the author taking $||x-y||<\epsilon$?

I'm not sure, but I believe so.

(2) What does the subscript $1_\{||X_n-Y_n||>\epsilon\}$ means and why is it there? I think the author divided the those values who were at a distance smaller than $\epsilon$ and those who weren't, but I'm not completely sure about this.

It's the characteristic function for that set. In this case, it evaluates to 1 on the set where $||X_n - Y_n|| > \epsilon$. The function evaluates to 0 on the complement of this set. See https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory).

(3) Why is it $E\{|f(X_n)-f(Y)n|1_{||X_n-Y_n||>\epsilon}\}\leq 2\alpha P\{||X_n-Y_n||\}$? This might be related to my second question, I am not completely sure about the meaning of the subscript. I see $f$ is bounded by $\alpha$, but why is it $2\alpha$?

My guess: We have $|f(x)| < \alpha$ due to the Lipschitz continuity, so $|f(x) - f(y)| < 2\alpha$.

BTW, don't surround things with $ signs -- they don't do anything on this site. Use a pair of $$ markers for standalone TeX, and a pair of ## markers for inline TeX.

 

[GabrielN00]

I'm not sure, but I believe so.
It's the characteristic function for that set. In this case, it evaluates to 1 on the set where $||X_n - Y_n|| > \epsilon$. The function evaluates to 0 on the complement of this set. See https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory).

My guess: We have $|f(x)| < \alpha$ due to the Lipschitz continuity, so $|f(x) - f(y)| < 2\alpha$.

BTW, don't surround things with $ signs -- they don't do anything on this site. Use a pair of $$ markers for standalone TeX, and a pair of ## markers for inline TeX.

Thank you. In regard to the comment about Lipschitz continuity, isn't $k$ de Lipschitz constant? And what kind of inequality justified the change from expectation to probability in (3)?

 

[Mark44]

Thank you. In regard to the comment about Lipschitz continuity, isn't $k$ de Lipschitz constant? And what kind of inequality justified the change from expectation to probability in (3)?

I don't know, but take a look here -- https://en.wikipedia.org/wiki/Slutsky's_theorem -- for an outline of the proof

 

[Ray Vickson]

Homework Statement

View attachment 213380

2. Questions

I want to know how this proof works, but there are several things I don't understand.

(1) How does the $k\epsilon$ appears? $k$ is the lipschitz constant, but what about $\epsilon$? Is the author taking $||x-y||<\epsilon$?

(2) What does the subscript $1_\{||X_n-Y_n||>\epsilon\}$ means and why is it there? I think the author divided the those values who were at a distance smaller than $\epsilon$ and those who weren't, but I'm not completely sure about this.

(3) Why is it $E\{|f(X_n)-f(Y)n|1_{||X_n-Y_n||>\epsilon}\}\leq 2\alpha P\{||X_n-Y_n||\}$? This might be related to my second question, I am not completely sure about the meaning of the subscript. I see $f$ is bounded by $\alpha$, but why is it $2\alpha$?

The author splits up $E|f(X_n)-f(Y_n)|$ into two parts: (i) the part where $|X_n - Y_n| < \epsilon$ (so that the expectation on that part is $< \epsilon k$ because we have $|f(X_n) - f(Y_n) | \leq k |X_n - Y_n| < k \epsilon$); and (ii) the part where $|X_n - Y_n | > \epsilon$, over which we have $|f(X_n) - f(Y_n)| \leq |f(X_n)| + |f(Y_n)| \leq 2 \alpha$, so that the expectation on that part is $\leq 2 \alpha$ times the probability of that part.

 

[StoneTemplePython]

For avoidance of doubt and confusion: do not call that a "characteristic function".

For probability purposes: things like $\mathbf 1_{\{X \leq x\}}$ or $\mathbb I_{\{X \leq x\}}$ are referred to as Indicator Functions.

In most branches of math we could call this a characteristic function -- not so in probability (and stats). In probability a 'characteristic function' is a reserved term and only used to refer to a kind of transform (basically a Fourier transform). If you look at the link provided by @Mark44 you can verify this for yourself.