arXiv subjects

Michael J. Klass

Publications and source records attributed to Michael J. Klass.

4 recordsLinked to original sources

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 $b=1/\alpha$, 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

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↗

arXiv2007

Pseudo-maximization and self-normalized processes

Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and parameter estimation, and Studentized pivots and bootstrap-$t$ methods for confidence intervals. In contrast to standard normalization, large values of the observations play a lesser role as they appear both in the numerator and its self-normalized denominator, thereby making the process scale invariant and contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of dependent variables and describe a key method called ``pseudo-maximization'' that has been used to derive these results. In the multivariate case, self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case) and is ubiquitous in statistical applications, examples of which are given.

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↗