arXiv subjects

Victor H. de la Pena

Publications and source records attributed to Victor H. de la Pena.

8 recordsLinked to original sources

arXiv2026

Sharp Decoupling Inequalities for the Variances and Second Moments of Sums of Dependent Random Variables

Both complete decoupling and tangent decoupling are classical tools aiming to compare two random processes where one has a weaker dependence structure. We give a new proof for the complete decoupling inequality, which provides a lower bound for the sum of dependent square-integrable nonnegative random variables $\sum\limits^n_{i=1} d_i$ \[ \frac{1}{2} \mathbb E \left( \sum\limits^n_{i=1} z_i \right)^2 \leq \mathbb E \left( \sum\limits^n_{i=1} d_i \right)^2, \] where $z_i \stackrel{\mathcal{L}}{=} d_i$ for all $i\leq n$ and $z_i$'s are mutually independent. We will then provide the following sharp tangent decoupling inequalities \[\mathbb Var \left( \sum\limits^n_{i=1} d_i\right) \leq 2 \mathbb Var \left( \sum\limits^n_{i=1} e_i\right),\] and \[\mathbb E \left( \sum\limits^n_{i=1} d_i\right)^2 \leq 2 \mathbb E \left( \sum\limits^n_{i=1} e_i\right)^2 - \left[ \mathbb E \left( \sum\limits^n_{i=1} e_i\right) \right]^2,\] where $\{e_i\}$ is the decoupled sequences of $\{d_i\}$ and $d_i$'s are not forced to be nonnegative. Applications to construct Chebyshev-type inequality and Paley-Zygmund-type inequality, and to bound the second moments of randomly stopped sums will be provided.

math.PR↗

arXiv2026

Beyond Wald's Equation and the Optional Sampling Theorem

This paper establishes a conservation identity for mean-zero martingales stopped by extended-valued stopping times. For any mean-zero martingale $\{M_n\}$ and any extended-valued stopping time $T$ satisfying $E|M_T|I(T<\infty)<\infty$, the quantity $L\equiv E[M_T I(T<\infty)]$ exists and equals $\lim_n E[-M_n I(T>n)]$, a limit which always exists. The optional sampling theorem for stopping times and uniformly integrable martingales -- and Wald's equation for mean-zero random variables, as its i.i.d.\ specialization -- is recovered with a little extra effort, in which case the limit also vanishes. The identity itself remains in force whether or not $L=0$, and whether or not $P(T<\infty)=1$. Two corollaries and an application derived from this identity provide information on the rate of decay of the tail probability of the stopping time. Moreover, a necessary and sufficient condition is presented to characterize when $E|M_T|I(T<\infty)$ is finite. The characterization applies more generally whenever $|M_n|$ is a sequence of random variables, each having finite expectation. A third theorem provides sufficient conditions ensuring that certain exceedance-level, potentially extended-valued, stopping times are finite with probability one. It further implies that $\limsup M_n=\infty$ almost surely. We demonstrate these results through examples and explore their implications for different families of martingales. Our findings extend classical results in martingale theory and provide new insights into the behavior of stopped martingales, especially when the expected value of the stopped martingale on the set where the extended-valued stopping time $T$ is finite differs from the expected value of the martingale at time 1.

math.PR↗

arXiv2026

Mitigating the Winner's Curse While Controlling Multiplicity: e-Process Methods for Anytime-Valid Inference in Dose-Ranging Trials

Phase II dose-ranging trials often report the largest observed dose-control effect while inspecting accumulating data repeatedly. This creates two coupled distortions: selection optimism from choosing the empirical winner, known as the winner's curse, and Type I error inflation from multiplicity across doses and interim looks. We develop an anytime-valid procedure for testing whether the best true dose effect exceeds a clinically meaningful margin. The mathematical starting point is a recent selection-premium identity for the running maximum: for dose-control scores, the expected gain from re-selecting the current leader becomes a predictable selection charge. Subtracting this charge gives a residual with nonpositive drift under the composite null; applying a one-sided mixture-exponential construction then yields an e-process and hence an anytime-valid global test. The resulting rule has a transparent ledger form: raw best effect minus selection charge minus monitoring margin, and a ``GO'' decision is made only when the remaining evidence still exceeds the clinical margin. We give plug-in implementations for Gaussian and binary outcomes, prove finite-sample Type I control and anytime lower confidence bounds for the best dose effect, and illustrate the method through a worked example and simulations.

stat.ME↗

arXiv2026

The Exact Ville Identity: From the Absorbing Case to the General Law with an Application to E-Values

For a nonnegative supermartingale $(M_n)$ with $M_0=1$, let $T_b$ be the first time it reaches a level $b>1$. Ville's inequality gives only the bound $\mathbb P(T_b<\infty)\le 1/b$. We prove the exact identity $\mathbb P(T_b<\infty)=(1-D_b-R_b)/(b+O_b)$, where $O_b$ is the expected overshoot at crossing, $D_b$ is the cumulative predictable supermartingale loss before crossing, and $R_b=\lim_n\mathbb E[M_n{\bf 1}\{T_b>n\}]$ is the residual mass carried by paths that never cross. Thus the slack in Ville's inequality is decomposed completely into overshoot, loss, and survival. The proof is given first in the absorbing case, where non-crossing paths decay to zero and $R_b=0$, and then in full generality using a conservation identity for stopped martingales at extended-valued stopping times. The formula yields Ville's inequality as a corollary, gives a sharp tightness criterion, and is worked out in examples including double-or-absorb processes, multiplicative decay, gambler's ruin, likelihood-ratio martingales, and a bounded martingale with $R_b>0$. Finally, with $b=1/α$, the same identity gives the exact type-I error of a sequential e-value test and identifies when threshold recalibration can safely recover unused significance.

math.PR↗

arXiv2026

Bounded Difference Concentration for Infinitely Exchangeable Sequences with Applications to AI Benchmark Uncertainty

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants $c_1, \dots, c_n$ decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is $σ_{\mathrm{mix}}^2$-subgaussian, we establish a concentration inequality with an effective variance proxy of $\frac{1}{4}\sum_i c_i^2 + σ_{\mathrm{mix}}^2$. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.

stat.ML↗

arXiv2026

A Selection Premium Decomposition for the Expected Maximum of Random Walks

When $K$ models are evaluated on the same validation set of size $n$, the selected winner's apparent performance is biased upward. Suppose $K$ models are evaluated on a shared sequence of i.i.d. observations $X_1,\dots, X_n$, where model $k$ achieves response $f_k(X_i)$ with mean $μ_k = \mathbb E[f_k(X)]$. Writing $Y_{i,k} = f_k(X_i)-μ_k$ for the centered increment and $S_{n,k} = \sum_{i=1}^n Y_{i,k}$ for the centered cumulative score, the expected maximum satisfies $0\le\mathbb E\bigl[\max_k S_{n,k}\bigr] = \sum_{i=1}^n \mathbb E\bigl[φ_K(S_{i-1})\bigr]$ where $φ_K(u) = \mathbb{E}\bigl[\max_k(u_k + Y_k)\bigr] - \max_k u_k$, $u\in \mathbb R^K$, is the selection premium function. This formula corresponds to the null hypothesis case (all models are equal in the sense that they have the same mean), which clarifies that the bias arises from selection. While this decomposition follows from elementary conditioning and telescoping, we develop the analytical consequences in five directions. (i) structural properties of $φ_K$; (ii) extension to stopping times, recovering Wald's equation at $K=1$; (iii) a winner's curse decomposition for heterogeneous means; (iv) a universal bias concentration law showing that the first $α$-fraction of observations generates a $\sqrtα$-fraction of total bias.

math.ST↗

arXiv2022

Sharp Concentration Results for Heavy-Tailed Distributions

We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy $\mathbb{P}(X>t) \leq {\rm e}^{- I(t)}$, where $I: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing function and $I(t)/t \rightarrow α\in [0, \infty)$ as $t \rightarrow \infty$. Our main theorem can not only recover some of the existing results, such as the concentration of the sum of subWeibull random variables, but it can also produce new results for the sum of random variables with heavier tails. We show that the concentration inequalities we obtain are sharp enough to offer large deviation results for the sums of independent random variables as well. Our analyses which are based on standard truncation arguments simplify, unify and generalize the existing results on the concentration and large deviation of heavy-tailed random variables.

math.PR↗

arXiv2004

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

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several exponential and moment inequalities, particularly those related to laws of the iterated logarithm, for self-normalized random variables including martingales. Tail probability bounds are also derived. For random variables B_t>0 and A_t, let Y_t(λ)=\exp{λA_t-λ^2B_t^2/2}. We develop inequalities for the moments of A_t/B_{t} or sup_{t\geq 0}A_t/{B_t(\log \log B_{t})^{1/2}} and variants thereof, when EY_t(λ)\leq 1 or when Y_t(λ) is a supermartingale, for all λbelonging to some interval. Our results are valid for a wide class of random processes including continuous martingales with A_t=M_t and B_t=\sqrt < M>_t, and sums of conditionally symmetric variables d_i with A_t=\sum_{i=1}^td_i and B_t=\sqrt\sum_{i=1}^td_i^2. A sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in R^m, m\ge 1, is also established. Another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self-normalized by the square root of the sum of squares. The key ingredient in this development is a new exponential supermartingale involving \sum_{i=1}^td_i and \sum_{i=1}^td_i^2.

math.PR↗