Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws

Peña, Víctor H. de la; Klass, Michael J.; Lai, Tze Leung

doi:10.1214/009117904000000397

Cited by 54 publications

(38 citation statements)

References 26 publications

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“…The literature on self-normalized bounds makes extensive use of the method of mixtures, sometimes called pseudo-maximization [15][16][17][18]20]; these works introduced the idea of using a mixture to bound a quantity with a random intrinsic time V t . These results are mostly given for fixed samples or finite time horizon, though de la Peña, Klass and Lai [15], equation (4.20), includes an infinite-horizon curve-crossing bound. Lai [41] treats confidence sequences for the parameter of an exponential family using mixture techniques similar to those of Section 3.2.…”

mentioning

confidence: 99%

Time-uniform, nonparametric, nonasymptotic confidence sequences

et al. 2021

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A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cramér-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL) and the sequential probability ratio test-our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.

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confidence: 99%

Time-uniform, nonparametric, nonasymptotic confidence sequences

et al. 2021

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“…(the additional factor e on the right-hand side is introduced artificially to encompass all t ≥ 0, also those for which p < 1; note that in this case the right-hand side exceeds one). We remark that similar self-normalized inequalities are known for example, in the theory of empirical processes (see [12]). The lower tail inequalities give…”

Section: Concentration Inequalities For General Convex Functionsmentioning

confidence: 57%

On the convex Poincaré inequality and weak transportation inequalities

Adamczak¹,

Strzelecki²

2019

Bernoulli

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We prove that for a probability measure on R n , the Poincaré inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line.The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest.We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar, and Tetali.

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“…Observe that this result is a consequence of more general bounds on self-normalized processes of the form X t = A t /B t (e.g. [8]), where in this case A t = M t is a martingale and B 2 t − 1 = M t its quadratic variation.…”

Section: Controlling W N T (H) Via a Maximal Inequality For Self-normalized Processesmentioning

confidence: 89%

A law of large numbers for interacting diffusions via a mild formulation

Bechtold¹,

Coppini

2021

Electron. J. Probab.

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Consider a system of n weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called McKean-Vlasov or Fokker-Planck equation, as n tends to infinity. We propose a relatively new approach to show this convergence by directly studying the stochastic partial differential equation that the empirical measure satisfies for each fixed n. Under a suitable control on the noise term, which appears due to the finiteness of the system, we are able to prove that the stochastic perturbation goes to zero, showing that the limiting measure is a solution to the classical McKean-Vlasov equation. In contrast with known results, we do not require any independence or finite moment assumption on the initial condition, but the only weak convergence.The evolution of the empirical measure is studied in a suitable class of Hilbert spaces where the noise term is controlled using two distinct but complementary techniques: rough paths theory and maximal inequalities for self-normalized processes.

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Self-normalized processes: exponential inequalities, moment bounds and iterated logarithm laws

Cited by 54 publications

References 26 publications

Time-uniform, nonparametric, nonasymptotic confidence sequences

Time-uniform, nonparametric, nonasymptotic confidence sequences

On the convex Poincaré inequality and weak transportation inequalities

A law of large numbers for interacting diffusions via a mild formulation

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