# An Infinitesimal Approach to Stochastic Analysis by H. Jerome Keisler

By H. Jerome Keisler

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**Extra resources for An Infinitesimal Approach to Stochastic Analysis**

**Sample text**

Fix A > 0. On the event {ξk > −A}, we have ξk+ ≤ ξk + A. Thus, for x ≥ 0, P{ξ1+ + ξ2+ > x} ≤ P{ξ1 + ξ2 > x − 2A, ξ1 > −A, ξ2 > −A} +P{ξ2 > x, ξ1 ≤ −A} + P{ξ1 > x, ξ2 ≤ −A} ≤ P{ξ1 + ξ2 > x − 2A} + 2F(x)F(−A). Hence, since F is long-tailed, lim sup x→∞ P{ξ1+ + ξ2+ > x} P{ξ1 + ξ2 > x − 2A} ≤ lim + 2F(−A) x→∞ F(x) F(x − 2A) = 2 + 2F(−A). Since A can be chosen as large as we please, lim sup x→∞ P{ξ1+ + ξ2+ > x} ≤ 2. e. that the distribution F + of ξ + is subexponential. 2 Subexponential Distributions on the Whole Real Line 43 (ii)⇒(i).

I)⇒(ii). Suppose that F is long-tailed. Fix A > 0. On the event {ξk > −A}, we have ξk+ ≤ ξk + A. Thus, for x ≥ 0, P{ξ1+ + ξ2+ > x} ≤ P{ξ1 + ξ2 > x − 2A, ξ1 > −A, ξ2 > −A} +P{ξ2 > x, ξ1 ≤ −A} + P{ξ1 > x, ξ2 ≤ −A} ≤ P{ξ1 + ξ2 > x − 2A} + 2F(x)F(−A). Hence, since F is long-tailed, lim sup x→∞ P{ξ1+ + ξ2+ > x} P{ξ1 + ξ2 > x − 2A} ≤ lim + 2F(−A) x→∞ F(x) F(x − 2A) = 2 + 2F(−A). Since A can be chosen as large as we please, lim sup x→∞ P{ξ1+ + ξ2+ > x} ≤ 2. e. that the distribution F + of ξ + is subexponential.

Hence F ∗ G(x) ∼ F ∗ G(x + y) as x → ∞. 41 since in each case the measure F + G is long-tailed. 42. Let the distributions F and G be long-tailed. Then the convolution F ∗ G is also long-tailed. 43. Suppose that F and G are distributions and that F is long-tailed. Suppose also that G(x) = o(F(x)) as x → ∞. Then F ∗ G is long-tailed. Finally in this section we have the following converse result. 44. Let F and G be two distributions on R+ such that F has unbounded support and G is non-degenerate at 0.