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Exponential inequalities for self-normalized processes with applications. (English) Zbl 1189.60042

Summary: We prove the following exponential inequality for a pair of random variables \((A,B)\) with \(B >0\) satisfying the canonical assumption, \(E[\exp(\lambda A - \frac{\lambda^2}{2}\;B^2)] \leq 1\) for \(\lambda\in \mathbb R\),
\[ P\left(\frac{|A|}{\sqrt{\frac{2q-1}{q} (B^2+ (E[|A|^p])^{2/p})}} \geq x \right) \leq \left(\frac{q}{2q-1}\right)^{\frac{q}{2q-1}} x^{-\frac{q}{2q-1}} e^{-x^2/2} \]
for \(x>0\), where \(1/p+ 1/q =1\) and \(p\geq 1\). Applying this inequality, we obtain exponential bounds for the tail probabilities for self-normalized martingale difference sequences. We propose a method of hypothesis testing for the \(L^p\)-norm \((p \geq 1)\) of \(A\) (in particular, martingales) and some stopping times. We apply this inequality to the stochastic TSP in \([0,1]^d (d\geq 2)\), connected to the CLT.

MSC:

60E15 Inequalities; stochastic orderings
60G42 Martingales with discrete parameter
62F03 Parametric hypothesis testing
62L15 Optimal stopping in statistics
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