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 k n and S n, k /V k n where \(U_n^2 = \sum\limits_{j = 1}^{m_n } {X_{n j}^2 }\) and V 2 n V 2 n is the conditional variance.
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Research supported by the National Science Foundation under Grant DMS-8601346
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Teicher, H. Distribution and moment convergence of martingales. Probab. Th. Rel. Fields 79, 303–316 (1988). https://doi.org/10.1007/BF00320924
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DOI: https://doi.org/10.1007/BF00320924
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Normal Variable
- Conditional Variance