In this paper we present an upper bound for the tail of the distribution of the supremum (taken for t in a fixed set K) of the modulus of expressions of the form 

$$G(t)=\sum_{n=1}^{n}a_{n}(e^{iX_{n}t}-E[e^{iX_{n}t}])$$

where X 1, X 2,…, X N are independent random variables each of which assumes values in some bounded interval [—L, L], a 1,a 2,…,a n are real or complex constants, N is a fixed positive integer, t is a real number, and K is some interval in ℜ1 of Lebesgue measure ∣K∣. Our specific interest in this problem is in obtaining explicit and mathematically rigorous numerical bounds for probabilities of the above type, with the goal of applying our results to response patterns that arise from arrays of randomly placed sensing devices. Thus, in addition to providing the probabilistic bounds described above we devote part of this paper to an explanation of a particular setting to which they apply. We comment at the outset that the problem of the distribution of the supremum of expressions (1.1) has been investigated in both the mathematical and the engineering literature, but that the two disciplines have had somewhat different research objectives and methods. These separate histories are traced briefly below, for there has been relatively little interplay between them.

Mathematical Analysis of Random Noise-Conclusion

Tail estimates for empirical characteristic functions, with applications to random arrays

We begin by describing in an informal fashion the subject matter of this chapter. The part of the general theory of stochastic processes dealing with countable sets of points randomly distributed on the real line or in an arbitrary space (for instance, Cartesian d-dimensional space) is called the “Theory of Point Processes.” Of all point processes, those on the real line have been most widely studied. Notwithstanding their relatively simple structure, they form building blocks in a variety of industrial, biological, geophysical, and engineering applications. The following example describes a general situation which in a natural fashion introduces a point process on a line.

The Poisson Process and Its Ramifications

A scan statistic is examined for the purpose of testing the existence of a global peak in a random process with dependent variables of any distribution. The scan statistic tail probability is obtained based on the covariance of the moving sums process, thereby accounting for the spatial nature of the data as well as the size of the searching window. Exact formulas linking this covariance to the window size and the correlation coefficient are developed under general, common and auto covariance structures of the variables in the original process. The implementation and applicability of the formulas are demonstrated on multiple processes of t-statistics, treating also the case of unknown covariance. A sensitivity analysis provides further insight into the variant interaction of the tail probability with the influence parameters. An R code for the tail probability computation and the data analysis is offered within the supplementary material.

Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size

Some properties of a gaussian random function between known zeros

https://www.jstage.jst.go.jp/article/ieejeiss1972/95/11/95_11_253/_pdf

Mathematical Analysis of Random Noise-Conclusion

Citations

Tail estimates for empirical characteristic functions, with applications to random arrays

The Poisson Process and Its Ramifications

Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size

Some properties of a gaussian random function between known zeros

Effect of non-white references on the statistical accuracy of a random reference correlator

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