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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?

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    $\begingroup$ Erm... $EY_n^\alpha\le EX_n^\alpha EX_1^{-\alpha}<+\infty$ (unless I am misreading something in your post). $\endgroup$
    – fedja
    Commented Oct 14, 2013 at 9:01

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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 $1$.

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