From moment growth to a tail bound

ProofAdvanced⏱ 1 hourHigh-Dim Probabilitymomentstailsequivalence

Problem

Suppose $\|X\|_{L^p}\le K\sqrt p$ for all $p\ge1$ . Prove directly (without going through the MGF) that

\P(|X|\ge t)\le e^{-ct^2/K^2}

for an absolute constant $c>0$ and all $t\ge0$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

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