Summary
Let F n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define \(\bar \mu _n^* (u) = \sqrt n [F_n (u) - u]\), and consider the integrals \(I(t) = \int\limits_0^t {\int\limits_0^1 {...\int\limits_0^1 {f(u_1 ,...,u_s )} } }\) where f is a bounded measurable function. We give a good upper bound on the probability \(P{\text{ }}(\mathop {\sup }\limits_{0 \leqq t \leqq 1} {\text{ | }}I{\text{(}}t{\text{) |}} \geqq x)\). An analogous estimate is given for multiple integrals with respect to a Poisson process.
References
Major, P., Rejtö, L.: Strong embedding of estimation of the distribution function under random censorship. Ann. Statist., 15 (1988)
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Major, P. On the tail behaviour of the distribution function of multiple stochastic integrals. Probab. Th. Rel. Fields 78, 419–435 (1988). https://doi.org/10.1007/BF00334204
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DOI: https://doi.org/10.1007/BF00334204
Keywords
- Distribution Function
- Uniform Distribution
- Stochastic Process
- Probability Theory
- Measurable Function