scholarly journals Limit theorems for multifractal products of geometric stationary processes

Bernoulli ◽  
2016 ◽  
Vol 22 (4) ◽  
pp. 2579-2608 ◽  
Author(s):  
Denis Denisov ◽  
Nikolai Leonenko
Keyword(s):  
Limit Theorems ◽  
Stationary Processes

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (01) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Fractional integrals of stationary processes and the central limit theorem

1976 ◽  
Vol 13 (4) ◽  
pp. 723-732 ◽  
Author(s):  
M. Rosenblatt
Keyword(s):  
Limit Theorems ◽  
Fractional Integral ◽  
Burgers Equation ◽  
Fractional Integrals ◽  
Stationary Processes ◽  
Singular Behavior ◽  
Mixing Processes ◽  
The Central Limit Theorem ◽  
Strongly Mixing ◽  
Strongly Mixing Processes

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.

Fractional integrals of stationary processes and the central limit theorem

1976 ◽  
Vol 13 (04) ◽  
pp. 723-732 ◽  
Author(s):  
M. Rosenblatt
Keyword(s):  
Limit Theorems ◽  
Fractional Integral ◽  
Burgers Equation ◽  
Fractional Integrals ◽  
Stationary Processes ◽  
Singular Behavior ◽  
Mixing Processes ◽  
The Central Limit Theorem ◽  
Strongly Mixing ◽  
Strongly Mixing Processes

A class of limit theorems involving asymptotic normality is derived for stationary processes whose spectral density has a singular behavior near frequency zero. Generally these processes have ‘long-range dependence’ but are generated from strongly mixing processes by a fractional integral or derivative transformation. Some related remarks are made about random solutions of the Burgers equation.

Gordin's theorem and the periodogram

1976 ◽  
Vol 13 (02) ◽  
pp. 365-370 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Stationary Sequence ◽  
Central Limit Theorems ◽  
Stationary Processes

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

scholarly journals Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

2019 ◽  
Vol 23 ◽  
pp. 803-822
Author(s):  
Mikkel Slot Nielsen ◽  
Jan Pedersen
Keyword(s):  
Quadratic Form ◽  
Continuous Time ◽  
Quadratic Forms ◽  
Statistical Estimation ◽  
Limit Theorems ◽  
Stationary Process ◽  
Moving Average ◽  
Sufficient Conditions ◽  
Stationary Processes ◽  
Limiting Behavior

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

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