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I need help to prove the following bound. I try to apply more well-known probabilistic inequality,but they don't work, and I don't know where to start.

Let $(X_n)$ be nonnegative i.i.d. r.v. such that $0<\mathbb{E}(X_1)\leq \infty$. Show that for any $t<\mathbb{E}(X_1)$, there is a positive constant $K$ depending on $t$ such that $\mathbb{P}\left( \frac{1}{n} \sum_{i=1}^n X_i<t\right) \leq \exp(-Kn)$ for all large enough $n$.


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The usual exponential bound based on Markov inequality works. What is stopping you? – Did Nov 4 '12 at 16:54
@did: I am not sure if this is correct, but what I tried is to use argument similar to that in proving Chernoff bound. If $E(X_1)$ is finite, then by strong law of large number the average is near to $E(X_1)$ a.s. for $n$ large enough, and then I use Hoeffding lemma to get bound for $E(e^{tX_i})$ and then proceed as usual. If $E(X_1)=\infty$, I am not sure how to proceed. – digiboy1 Nov 5 '12 at 16:12
Sure "this is correct", now? – Did Oct 16 '13 at 4:41
up vote 2 down vote accepted

This is to show that the non integrable case follows from the integrable one. Assume that for every i.i.d. integrable $(X_n)_{n\geqslant1}$ and every $t\lt\mathbb E(X_1)$, there exists some positive $K$, depending on $t$ and the distribution of $X_1$ but not on $n$, such that $\mathbb P(A_n(t))\leqslant\mathrm e^{-Kn}$ for every $n$, where $$ A_n(t)=[X_1+\cdots+X_n\lt nt]. $$ Consider now some nonintegrable nonnegative i.i.d. sequence $(X_n)_{n\geqslant1}$ and some finite $t$. For every $x\gt0$, consider $X_n^x=\min(X_n,x)$. Since $\mathbb E(X_1^x)\to+\infty$ when $x\to+\infty$, there exists some $x$ such that $\mathbb E(X_1^x)\gt t$. Then $$ A_n(t)\subseteq A_n^x(t)=[X^x_1+\cdots+X^x_n\lt nt], $$ and, since $X_1^x$ is integrable, by the result in the integrable case, $\mathbb P(A_n(t))\leqslant\mathbb P(A_n^x(t))\leqslant\mathrm e^{-Kn}$ for every $n$, for the positive $K$ corresponding to $t$ and the distribution of $X_1^x$.

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