ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary If $$S_{n,k} = \mathop \Sigma \limits_{1 \leqq i_1 〈 i_k \leqq m_n } X_{ni_1 } ...{\text{ }}X_{ni_k } $$ where {X n j ,ℱ n j 1≦j≦m n ↑∞, n≧1} is a martingale difference array, conditions are given for the distribution and moment convergence of S n,k to the distribution and moments of $$\frac{1}{{k!}}H_k (Z)$$ where H k is the Hermite polynomial of degree k and Z is a standard normal variable. This is intimately related to an identity (*) for multiple Wiener integrals. Under alternative conditions, similar results hold for S n, k /U n k and S n, k /V n k where $$U_n^2 = \sum\limits_{j = 1}^{m_n } {X_{n j}^2 }$$ and V n 2 V n 2 is the conditional variance.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00320924
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