Related papers: Prokhorov Metric Convergence of the Partial Sum Pr…
Samples of dynamic or time-varying networks and other random object data such as time-varying probability distributions are increasingly encountered in modern data analysis. Common methods for time-varying data such as functional data…
Irregular functional data in which densely sampled curves are observed over different ranges pose a challenge for modeling and inference, and sensitivity to outlier curves is a concern in applications. Motivated by applications in…
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…
The use of quadratic forms of the empirical process for the two-sample problem in the context of functional data is considered. The convergence of the family of statistics proposed to a Gaussian limit is established under metric entropy…
It has been observed by several authors that well-known periodization strategies like tent or Chebychev transforms lead to remarkable results for the recovery of multivariate functions from few samples. So far, theoretical guarantees are…
Under certain mild conditions, limit theorems for additive functionals of some $d$-dimensional self-similar Gaussian processes are obtained. These limit theorems work for general Gaussian processes including fractional Brownian motions,…
We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is…
The Skorokhod reflection was used in 1961 to create a reflected diffusion on the half-line. Later, it was used for processes with jumps such as reflected L\'evy processes. Like a Brownian motion, which is a weak limit of random walks,…
Donsker Theorem is perhaps the most famous invariance principle result for Markov processes. It states that when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general…
Statistically self-similar measures on $[0,1]$ are limit of multiplicative cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$. These weights are i.i.d, positive, and of expectation $1/b$. We extend these cascades…
We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian…
Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the…
The Wasserstein distance between two probability measures on a metric space is a measure of closeness with applications in statistics, probability, and machine learning. In this work, we consider the fundamental question of how quickly the…
The potential applications of boundary functionals of random processes, such as the extreme values of these processes, the moment of first reaching a fixed level, the value of the process at the moment of reaching the level, the moment of…
We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special…
We study a random process with reinforcement, which evolves following the dynamics of a given diffusion process in a bounded domain and is resampled according to its occupation measure when it reaches the boundary. We show that its…
The estimation of regression parameters in one dimensional broken stick models is a research area of statistics with an extensive literature. We are interested in extending such models by aiming to recover two or more intersecting…
The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for…
In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale…
Despite the success of fractional Brownian motion (fBm) in modeling systems that exhibit anomalous diffusion due to temporal correlations, recent experimental and theoretical studies highlight the necessity for a more comprehensive approach…