Related papers: Nonstandard Convergence Gives Bounds on Jumps
It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution…
Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates a class of random dynamical systems, arising from perturbing a one-dimensional piecewise…
A vast literature on convergence guarantees for gradient descent and derived methods exists at the moment. However, a simple practical situation remains unexplored: when a fixed step size is used, can we expect gradient descent to converge…
For non-uniformly hyperbolic dynamical systems we consider the time series of maxima along typical orbits. Using ideas based upon quantitative recurrence time statistics we prove convergence of the maxima (under suitable normalization) to…
Recent studies revealed complex convergence dynamics in gradient-based methods, which has been little understood so far. Changing the step size to balance between high convergence rate and small generalization error may not be sufficient:…
The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard…
Let $(X_i)_{i=1,...,n}$ be a possibly nonstationary sequence such that $\mathscr{L}(X_i)=P_n$ if $i\leq n\theta$ and $\mathscr{L}(X_i)=Q_n$ if $i>n\theta$, where $0<\theta <1$ is the location of the change-point to be estimated. We…
The Central Limit Theorem states that, in the limit of a large number of terms, an appropriately scaled sum of independent random variables yields another random variable whose probability distribution tends to a stable distribution. The…
In this paper, we are interested in testing if the volatility process is constant or not during a given time span by using high-frequency data with the presence of jumps and microstructure noise. Based on estimators of integrated volatility…
Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that…
We propose consensus propagation, an asynchronous distributed protocol for averaging numbers across a network. We establish convergence, characterize the convergence rate for regular graphs, and demonstrate that the protocol exhibits better…
For any recurrent random walk (S_n)_{n>0} on R, there are increasing sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at least one finite accumulation point. For one class of random walks, we give a criterion on…
In this paper, we show that there is a close relation between consistency in a constraint network and set intersection. A proof schema is provided as a generic way to obtain consistency properties from properties on set intersection. This…
We study random compositions of transformations having certain uniform fiberwise properties and prove bounds which in combination with other results yield a quenched central limit theorem equipped with a convergence rate, also in the…
Uniform deviation bounds limit the difference between a model's expected loss and its loss on an empirical sample uniformly for all models in a learning problem. As such, they are a critical component to empirical risk minimization. In this…
This paper shows that jumps in financial asset prices are often erroneously identified and are, in fact, rare events accounting for a very small proportion of the total price variation. We apply new econometric techniques to a comprehensive…
We prove a strong concentration result about the natural collision estimator, which counts the number of collisions that occur within an iid sample. This estimator is at the heart of algorithms used for uniformity testing and entropy…
Stable laws can be tempered by modifying the L\'evy measure to cool the probability of large jumps. Tempered stable laws retain their signature power law behavior at infinity, and infinite divisibility. This paper develops random walk…
We consider uniform moment convergence of lag-window spectral density estimates for univariate and multivariate stationary processes. Optimal rates of convergence are obtained under mild and easily verifiable conditions. Our theory…
Consequences of the basic and most evident consistency requirement-that measured events cannot happen and not happen at the same time-are shortly reviewed. Particular emphasis is given to event forecast and event control. As a consequence,…