scholarly journals The partial sum process of orthogonal expansions as geometric rough process with Fourier series as an example—An improvement of Menshov–Rademacher theorem

2013 ◽  
Vol 265 (12) ◽  
pp. 3067-3103 ◽  
Author(s):  
Terry J. Lyons ◽  
Danyu Yang
Keyword(s):  
Fourier Series ◽  
Orthogonal Expansions ◽  
Partial Sum ◽  
Partial Sum Process ◽  
Rademacher Theorem

The influence of sequential extremal processes on the partial sum process

Extremes ◽  
2012 ◽  
Vol 16 (1) ◽  
pp. 39-54
Author(s):  
Arnold Janssen ◽  
Markus Pauly
Keyword(s):  
Partial Sum ◽  
Partial Sum Process ◽  
Extremal Processes

scholarly journals THE INVARIANCE PRINCIPLE FOR LINEAR PROCESSES WITH APPLICATIONS

2002 ◽  
Vol 18 (1) ◽  
pp. 119-139 ◽  
Author(s):  
Qiying Wang ◽  
Yan-Xia Lin ◽  
Chandra M. Gulati
Keyword(s):  
Wiener Process ◽  
Invariance Principle ◽  
Random Variables ◽  
Linear Process ◽  
Standard Wiener Process ◽  
Test Statistic ◽  
Linear Processes ◽  
Partial Sum ◽  
Unit Root Testing ◽  
Partial Sum Process

Let Xt be a linear process defined by Xt = [sum ]k=0∞ ψkεt−k, where {ψk, k ≥ 0} is a sequence of real numbers and {εk, k = 0,±1,±2,...} is a sequence of random variables. Two basic results, on the invariance principle of the partial sum process of the Xt converging to a standard Wiener process on [0,1], are presented in this paper. In the first result, we assume that the innovations εk are independent and identically distributed random variables but do not restrict [sum ]k=0∞ |ψk| < ∞. We note that, for the partial sum process of the Xt converging to a standard Wiener process, the condition [sum ]k=0∞ |ψk| < ∞ or stronger conditions are commonly used in previous research. The second result is for the situation where the innovations εk form a martingale difference sequence. For this result, the commonly used assumption of equal variance of the innovations εk is weakened. We apply these general results to unit root testing. It turns out that the limit distributions of the Dickey–Fuller test statistic and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test statistic still hold for the more general models under very weak conditions.

On absolute summability factors of infinite series and their application to Fourier series

1967 ◽  
Vol 63 (1) ◽  
pp. 107-118 ◽  
Author(s):  
R. N. Mohapatra ◽  
G. Das ◽  
V. P. Srivastava
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Infinite Series ◽  
Absolute Summability ◽  
Summability Factors ◽  
Partial Sum ◽  
Absolute Summability Factors ◽  
Image Position

Definition. Let {sn} be the n-th partial sum of a given infinite series. If the transformationwhereis a sequence of bounded variation, we say that εanis summable |C, α|.

Some remarks on regression with autoregressive errors and their residual processes

1987 ◽  
Vol 24 (3) ◽  
pp. 668-678 ◽  
Author(s):  
R. J. Kulperger
Keyword(s):  
Linear Regression ◽  
Regression Models ◽  
Random Polynomial ◽  
Linear Regression Models ◽  
Partial Sum ◽  
Brownian Bridges ◽  
Partial Sum Process ◽  
Autoregressive Errors ◽  
Ar Process

We consider some linear regression models Y = Σα lfl(z) + X, where X is an autoregressive (AR) process. The residuals estimate the i.i.d. innovations sequence which drives the AR process. We then consider the partial sum process of the residuals and show they converge to Brownian bridges in certain cases. Some remarks are also made on similar processes when differencing is first applied to remove trends. When an AR process is differenced the residual partial sum can be asymptotically a random polynomial.

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