0
$\begingroup$

Let $X_n$ be a sequence of iid positive random variables. Assume that $X_n$ has finite $\alpha$th moment for some value $\alpha \in (0,1)$, but infinite first moment. Assume also that the reciprocal $1/X_n$ has finite moment generating function.

Define the random variable $$Y_n := \frac{X_n}{X_1 + \cdots + X_{n-1}}.$$ Is it possible for $Y_n$ to have infinite moments of all order? If so, could you provide a counterexample?

$\endgroup$
  • 3
    $\begingroup$ Erm... $EY_n^\alpha\le EX_n^\alpha EX_1^{-\alpha}<+\infty$ (unless I am misreading something in your post). $\endgroup$ – fedja Oct 14 '13 at 9:01
1
$\begingroup$

Since $1/X_j$ has a finite moment generating function, the random variable $\frac 1{X_1+\dots+X_{n-1}}$ has moments of any order. Using independence, we thus have that $Y_n\in\mathbb L^p$ if and only if $X_n\in\mathbb L^p$.

In particular, $Y_n$ has finite moments of order $\alpha$ but not of order $1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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