Measuring the magnitude of sums of independent random variables

测量信号的粗糙度

Measuring the roughness of a signal

本文提出基于归一化幂变差的非参数方法估计路径粗糙度，在Hölder正则等条件下证明一致性并给出收敛速度，适用于分数布朗运动等，并通过数值实验验证。

Dependency-dependent Bounds for Sums of Dependent Random Variables

We consider the problem of bounding large deviations for non-i.i.d. random variables that are allowed to have arbitrary dependencies. Previous works typically assumed a specific dependence structure, namely the existence of independent components. Bounds that depend on the degree of dependence between the observations have only been studied in the theory of mixing processes, where variables are time-ordered. Here, we introduce a new way of measuring dependences within an unordered set of variables. We prove concentration inequalities, that apply to any set of random variables, but benefit from…

On the Order of Magnitude of Sums of Negative Powers of Integrated Processes

The asymptotic behavior of expressions of the form $% \sum_{t=1}^{n}f(r_{n}x_{t})$ where $x_{t}$ is an integrated process, $r_{n}$ is a sequence of norming constants, and $f$ is a measurable function has been the subject of a number of articles in recent years. We mention Borodin and Ibragimov (1995), Park and Phillips (1999), de Jong (2004), Jeganathan (2004), Pötscher (2004), de Jong and Whang (2005), Berkes and Horvath (2006), and Christopeit (2009) which study weak convergence results for such expressions under various conditions on $x_{t}$ and the function $f$. Of course, these results al…

New Berry-Esseen and Wasserstein bounds in the CLT for non-randomly centered random sums by probabilistic methods

We prove abstract bounds on the Wasserstein and Kolmogorov distances between non-randomly centered random sums of real i.i.d. random variables with a finite third moment and the standard normal distribution. Except for the case of mean zero summands, these bounds involve a coupling of the summation index with its size biased distribution as was previously considered in \cite{GolRin96} for the normal approximation of nonnegative random variables. When being specialized to concrete distributions of the summation index like the Binomial, Poisson and Hypergeometric distribution, our bounds turn ou…

On the limit distributions of some sums of a random multiplicative function

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n \leq x with k distinct prime factors, provided that k = o(log log x) as x \rightarrow \infty. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log log x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n \leq x with a…

Prediction of components in random sums

We consider predictions of the random number and the magnitude of each iid component in a random sum based on its distributional structure, where only a total value of the sum is available and where iid random components are non-negative. The problem is motivated by prediction problems in a Poisson shot noise process. In the context, although conditional moments are best possible predictors under the mean square error, only a few special cases have been investigated because of numerical difficulties. We replace the prediction problem of the process with that of a random sum, which is more gene…