scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Moment Inequalities for the Maximum Cumulative Sum

01 Aug 1970-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 41, Iss: 4, pp 1227-1234
TL;DR: In this article, the inequality due to Rademacher-Mensov for orthogonal $X_i$'s is generalized to other types of dependent Rv's.
Abstract: Assume $E(X_i) \equiv 0$. For $ u \geqq 2$, bounds on the $ u$th moment of $\max_{1 \leqq k \leqq n}|\sum^{a + k}_{a + 1} X_i|$ are deduced from assumed bounds on the $ u$th moment of $|\sum^{a + n}_{a + 1} X_i|$. The inequality due to Rademacher-Mensov for $ u = 2$ and orthogonal $X_i$'s is generalized to $ u \geqq 2$ and other types of dependent $\operatorname{rv's}.$ In the case $ u > 2$, a second result is obtained which is considerably stronger than the first for asymptotic applications.

Content maybe subject to copyright    Report

Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, a new test statistic is derived for the hypothesis that a regression function has a prescribed parametric form, which is itself a smoothing parameter which is selected to minimize an estimated risk function.
Abstract: A new test is derived for the hypothesis that a regression function has a prescribed parametric form. Unlike many recent proposals, this test does not depend on arbitrarily chosen smoothing parameters. In fact, the test statistic is itself a smoothing parameter which is selected to minimize an estimated risk function. The exact distribution of the test statistic is obtained when the error terms in the regression model are Gaussian, while the large sample distribution is derived for more general settings. It is shown that the proposed test is consistent against fixed alternatives and can detect local alternatives that converge to the null hypothesis at the rate $1/\sqrt n$, where $n$ is the sample size. More importantly, the test is shown by example to have an ability to adapt to the alternative at hand.

216 citations

Journal ArticleDOI
TL;DR: In this paper, the convergence of rth(r>2) absolute moments in the central limit theorem for stationary φ-mixing and strong mixing sequences was studied for a strictly stationary strong mixing sequence.
Abstract: For an r>2 and a finite $$K,E\left| {\sum\limits_{j = 1}^n {X_j } } \right|^r \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } Kn^{r/2}$$ (all n≧1) is obtained for a strictly stationary strong mixing sequence {X j }. The convergence of rth(r>2) absolute moments in the central limit theorem for stationary φ-mixing and strong mixing sequences is also studied.

204 citations


Cites background from "Moment Inequalities for the Maximum..."

  • ...The following corollaries are due to Serfling ([ 13 ], Theorem B and [14],...

    [...]

Journal ArticleDOI
TL;DR: In this paper, strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds are established for linear and some nonlinear processes, and strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.
Abstract: We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.

197 citations

Journal ArticleDOI
TL;DR: In this paper, strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds are established for linear and some nonlinear processes, and strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.
Abstract: We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.

185 citations


Additional excerpts

  • ...Similar maximal inequalities appeared in the literature; see Menchoff [37], Doob [15], Billingsley [5], Serfling [50], Moricz [39] and Lai and Stout [33]....

    [...]