a b s t r a c tWe consider the equation R n = Q n + M n R n−1 , with random non-i.i.d. coefficients (Q n , M n ) n∈Z ∈ R 2 , and show that the distribution tails of the stationary solution to this equation are regularly varying at infinity.

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Abstract. If (dn) n≥0 is a martingale difference sequence, (εn) n≥0 a sequence of numbers in {1, −1}, and n a positive integer, thenHere αp denotes the best constant. If 1 ≤ p ≤ 2, then αp = 2/Γ(p + 1) as was shown by Burkholder. We show here that αp = p p−1 /2 for the case p > 2, and that p p−1 /2 is also the best constant in the analogous inequality for two martingales M and N indexed by [0, ∞), right continuous with limits from the left, adapted to the same filtration, and such that [M, M ]t −[N, N ]t is nonnegative and nondecreasing in t. In Section 7, we prove a similar inequality for harmonic functions.

ObjectiveThis study aimed to assess the association between bone mineral density (BMD) and coronary atherosclerosis in healthy postmenopausal women.MethodsWe performed a retrospective review of 252 postmenopausal women who had visited a health promotion center for a routine checkup. BMD of the lumbar spine (L1-L4) and femoral neck was evaluated using dual-energy X-ray absorptiometry, and coronary atherosclerosis was assessed using 64-row multidetector computed tomography. Participants were divided into normal BMD and osteopenia-osteoporosis groups, according to the T-scores of their lumbar spine or femoral neck.ResultsParticipants with osteopenia-osteoporosis had a significantly higher proportion of coronary atherosclerosis than did those with normal BMD at the lumbar spine (P=0.003) and femoral neck (P=0.004). Osteopenia-osteoporosis at the lumbar spine (odds ratio [OR], 2.86; 95% confidence interval [CI], 1.12 to 7.27) or femoral neck (OR, 3.35; 95% CI, 1.07 to 10.57) was associated with coronary atherosclerosis, after controlling for age and cardiovascular risk factors.ConclusionDecreased BMD is associated with coronary atherosclerosis in healthy postmenopausal women, independent of age and cardiovascular risk factors. Postmenopausal women with decreased BMD may have a higher risk of developing coronary atherosclerosis.