Asymptotic normality of quadratic forms of martingale differences

Liudas Giraitis*, Masanobu Taniguchi, Murad S. Taqqu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


We establish the asymptotic normality of a quadratic form Qn in martingale difference random variables ηt when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables ηt, asymptotic normality holds under condition | | A| | sp= o(| | A| |) , where | | A| | sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηt has been an important open problem. We provide such sufficient conditions in this paper.

Original languageEnglish
Pages (from-to)315-327
Number of pages13
JournalStatistical Inference for Stochastic Processes
Issue number3
Publication statusPublished - 2017 Oct 1
Externally publishedYes


  • Asymptotic normality
  • Martingale differences
  • Quadratic form

ASJC Scopus subject areas

  • Statistics and Probability


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