Optimal Gaussian Approximation For Multiple Time Series

Abstract

We obtain an optimal bound for a Gaussian approximation of a large class of vector-valued random processes. Our results provide a substantial generalization of earlier results that assume independence and/or stationarity. Based on the decay rate of the functional dependence measure, we quantify the error bound of the Gaussian approximation using the sample size n and the moment condition. Under the assumption of pth finite moment, with p > 2, this can range from a worst case rate of n 1/2 to the best case rate of n 1/p .

Time-varying auto-regressive models for count time-series

“…On the same note, even though we imposed the Poisson assumption for increased model interpretation, in the light of the upper bounds for the KL distance, it is not a necessary criterion and can be applied to a general multiple non-stationary count time-series. Extending some of the continuous time-series invariance results for nonlinear non-stationary and multiple series from [34] to a count series regime will be an interesting challenge. Finally, we wish to undertake an autoregressive estimation of the basic reproduction number with the time-varying version of compartmental models in epidemiology.…”

Sequential Gaussian approximation for nonstationary time series in high dimensions

“…Under mild regularity conditions, we also derive an explicit expression for the approximating Gaussian process in terms of local long run covariances, and suggest a multiplier bootstrap procedure to perform statistical inference. In contrast, the result of Karmakar and Wu (2020) does not provide an explicit expression for the Gaussian variance, while Berkes et al (2014) provide explicit expressions only for the stationary univariate case. Crucial for our feasible bootstrap approximation is some distributional regularity in time, which we formulate in terms of a total variation norm of the underlying kernel function.…”

“…We refer to Zaitsev (2013) for an extensive literature review on strong Gaussian approximations for independent random vectors. In the dependent case, this optimal rate has recently been achieved by Berkes et al (2014) for univariate stationary time series, and by Karmakar and Wu (2020) for multivariate nonstationary time series. Earlier results for multivariate time series are due to Liu and Lin (2009) for the stationary case, and Wu and Zhou (2011) for the nonstationary case.…”

Rates of convergence in invariance principles for random walks on linear groups via martingale methods

“…Let us mention that, to prove the ASIP for cocycles, a second way would be to apply a multidimensional version of the strong approximation result of Berkes, Liu and Wu [4], as given in Karmakar and Wu [22]. However, a direct application of this result would not give the good rate of convergence with respect to the moment of µ (see the introduction of [9] for more details).…”

“…However, a direct application of this result would not give the good rate of convergence with respect to the moment of µ (see the introduction of [9] for more details). Hence an adaptation of the result of [22], similar to what we did in [9] for real-valued cocycles, would be necessary here, and it is not clear wether this adaptation is feasible or not. Let us also mention that the ASIP for R d -valued martingales is interesting in itself, and can be useful in other situations.…”