By Mohamed Abdel-Hameed
This publication covers Lévy strategies and their functions within the contexts of reliability and garage. unique cognizance is paid to lifestyles distributions and the upkeep of units topic to degradation; estimating the parameters of the degradation strategy is additionally mentioned, as is the upkeep of dams topic to Lévy enter.
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Sample text
Let ↓ p(t, x) be the probability transition function of the process X . For any positive t, x we let ⎩∧ _ p(t, z)G(x + z)dz f (t, x) = 0 For any positive t, , we have ⎩∧ _ H (t + )= p(t + _ , x)G(x)d x 0 ⎩∧ = p( , x) f (t, x)d x, 0 ↓ where the last equality follows since X has stationary independent increments. From the last equality, it follows D( , t1 , t2 ) ≥ 0 if the determinant f t1 t2 x 0 ⎧ ⎧ ⎧ f (t1 , x) f (tx , x) ⎧ ⎧ ⎧ =⎧ f (t1 , 0) f (t2 , 0) ⎧ is positive. 5 we have f t1 t2 x 0 ⎧ _ ⎧⎧ _ ⎩⎩ ⎧ ⎧ p(t1 , z 1 ) p(t1 , z 2 ) ⎧ ⎧⎧ G(x + z 1 ) G(x + z 2 ) ⎧⎧ ⎧ ⎧ _ _ = ⎧.
For any positive t, x we let ⎩∧ _ p(t, z)G(x + z)dz f (t, x) = 0 For any positive t, , we have ⎩∧ _ H (t + )= p(t + _ , x)G(x)d x 0 ⎩∧ = p( , x) f (t, x)d x, 0 ↓ where the last equality follows since X has stationary independent increments. From the last equality, it follows D( , t1 , t2 ) ≥ 0 if the determinant f t1 t2 x 0 ⎧ ⎧ ⎧ f (t1 , x) f (tx , x) ⎧ ⎧ ⎧ =⎧ f (t1 , 0) f (t2 , 0) ⎧ is positive. 5 we have f t1 t2 x 0 ⎧ _ ⎧⎧ _ ⎩⎩ ⎧ ⎧ p(t1 , z 1 ) p(t1 , z 2 ) ⎧ ⎧⎧ G(x + z 1 ) G(x + z 2 ) ⎧⎧ ⎧ ⎧ _ _ = ⎧.
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