Asymptotic normality of quadratic forms of martingale differences

We establish asymptotic normality of weighted sums of linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and innovations. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular, they are applicable to GARCH and ARCH(∞) models and to their squares. They are also useful in deriving asymptotic normality of kernel-type estimators of a nonparametric regression function with short or long memory moving average errors.

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Quadratic Forms with Applications to Algebraic Geometry and Topology

10.1017/cbo9780511526077 ◽

1995 ◽

Cited By ~ 64

Author(s):

Albrecht Pfister

Keyword(s):

Algebraic Geometry ◽

Quadratic Forms ◽

And Topology

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Asymptotic normality of processes with a branching structure

Communication in Statistics- Theory and Methods ◽

10.1080/03610929808828951 ◽

1998 ◽

Vol 14 (4) ◽

pp. 833-848

Author(s):

Malcolm P. Quine ◽

Władysław Szczotka

Keyword(s):

Asymptotic Normality

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Key Recovery Attacks on Multivariate Public Key Cryptosystems Derived from Quadratic Forms over an Extension Field

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e100.a.18 ◽

2017 ◽

Vol E100.A (1) ◽

pp. 18-25 ◽

Cited By ~ 3

Author(s):

Yasufumi HASHIMOTO

Keyword(s):

Quadratic Forms ◽

Public Key ◽

Extension Field ◽

Key Recovery ◽

Public Key Cryptosystems ◽

Multivariate Public Key ◽

Key Recovery Attacks

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Existence

10.23943/princeton/9780691166902.003.0016 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Structure Theorem ◽

Quadratic Extension ◽

Quadratic Space ◽

Division Algebras ◽

Separable Extension ◽

Valued Fields ◽

Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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Totally Wild Quadratic Forms of Type E7

10.23943/princeton/9780691166902.003.0015 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Quadratic Space ◽

Trace Map ◽

Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

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Linked Tori, II

10.23943/princeton/9780691166902.003.0006 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Integral Domain ◽

Quadratic Space ◽

Small Observation ◽

Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

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