Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Central limit theorems revisited
AU - Majumdar, Suman
AU - Kundu, Subrata
AU - Mukherjee, Kanchan
PY - 2000/4/15
Y1 - 2000/4/15
N2 - A Central Limit Theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis. A Central Limit Theorem for real-valued martingale difference arrays is obtained under the conditional Lindeberg condition when the row sums of conditional variances converge to a (possibly degenerate) constant. This result is then extended, first to multi-dimensions and next to Hilbert-valued elements, under appropriate convergence requirements on the conditional and unconditional covariances and the conditional Lindeberg condition along (orthonormal) basis evaluations. Extension to include Banach- (with a Schauder basis) valued random elements is indicated.
AB - A Central Limit Theorem for a triangular array of row-wise independent Hilbert-valued random elements with finite second moment is proved under mild convergence requirements on the covariances of the row sums and the Lindeberg condition along the evaluations at an orthonormal basis. A Central Limit Theorem for real-valued martingale difference arrays is obtained under the conditional Lindeberg condition when the row sums of conditional variances converge to a (possibly degenerate) constant. This result is then extended, first to multi-dimensions and next to Hilbert-valued elements, under appropriate convergence requirements on the conditional and unconditional covariances and the conditional Lindeberg condition along (orthonormal) basis evaluations. Extension to include Banach- (with a Schauder basis) valued random elements is indicated.
KW - CLT
KW - Hilbert space
KW - Lindeberg condition
KW - Martingale dierence array
KW - Weak convergence
U2 - 10.1016/S0167-7152(99)00164-9
DO - 10.1016/S0167-7152(99)00164-9
M3 - Journal article
VL - 47
SP - 265
EP - 275
JO - Statistics and Probability Letters
JF - Statistics and Probability Letters
SN - 0167-7152
IS - 3
ER -