Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a question in my homework:

A continuous process $X$ is said to be self-similar if for every $\lambda>0$, $(X_{\lambda t})_{t\geq 0}$ has the same law as $(\lambda X_t)_{t\geq 0}$.

Let $X$ be self-similar and positive and for $p>1$, set

$$S_p=\sup_{s\geq 0}(X_s-s^p),\ \ X_t^{\ast}=\sup_{s\leq t}X_s$$

Prove there exists a constant $c_p$ depending only on $p$ such that for any $a>0$

$$P(c_p(X_t^{\ast})^p\geq a)\leq P(S_p\geq a)$$

Thanks a lot for your help

share|improve this question
    
Is $c_p$ allowed to also depend on $t$? –  Ben Derrett Oct 3 '12 at 16:25
    
Got something from the answer below? –  Did Sep 11 '13 at 16:48
    
There's a typo. The oringinal problem is to prove $P(c_p(X_1^{\ast})^p\geq a)\leq P(S_p\geq a)$ –  Danielsen Jun 30 at 14:47
add comment

1 Answer 1

The claimed result is doubtful: assume that $X_t=t$ for every $t$, almost surely. Then, $(X_t)_t$ is indeed self-similar, $\mathbb P(S_p\geqslant a)=0$ for every $a$ large enough while $\mathbb P(c\cdot(X^*_t)^p\geqslant a)=1$ for $t$ large enough, for every fixed positive $c$ and $a$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.