De La Peña, Victor H.; Pang, Guodong Exponential inequalities for self-normalized processes with applications. (English) Zbl 1189.60042 Electron. Commun. Probab. 14, 372-381 (2009). 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. Cited in 4 Documents MSC: 60E15 Inequalities; stochastic orderings 60G42 Martingales with discrete parameter 62F03 Parametric hypothesis testing 62L15 Optimal stopping in statistics Keywords:self-normalization; exponential inequalities; martingales; hypothesis testing; stochastic traveling salesman problem PDFBibTeX XMLCite \textit{V. H. De La Peña} and \textit{G. Pang}, Electron. Commun. Probab. 14, 372--381 (2009; Zbl 1189.60042) Full Text: DOI EuDML EMIS