Options, Futures, and other Derivatives(ISBN 97801350009949) by John C. Hull

By John C. Hull

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Additional info for Options, Futures, and other Derivatives(ISBN 97801350009949)

Example text

Proo/ 0/ Theorem 1. Let n P{m:xITkl2: R} = P(A) = LP(Ak). k=l Since the variables (i with i hypothesis that 6, ... >" k do not occur in the definition of Ak, the ,en are independent implies n E(T 2 ;Ak) = E(T;;Ak) + L E((l;Ak) 2: E(T;;Ak) 2: R 2 p(Ak), i=k-1 hence ET 2 2: R 2 P(A), which proves the desired inequality. 17 ON SUMS OF INDEPENDENT RANDOM VARIABLES Remark. This inequality is an identity, for example, in the following case: n = 1 and Proo/ 0/ Theorem 2. We set A;k = {11J I< iR, j< k; ITkl and ~ iR} n Ai = LAik.

This formula immediately implies the required condition. If the random variables 1]n have finite expectation, I: IxldF(x) < 00, then the condition of Theorem 12 holds, since then It can be shown that in this case we have normal stability. Thus the following theorem holds: Theorem 13. The stability ofthe means O'n = (1]1 + ... + 1]n)/n, n ~ 1, of a sequenee of independent identieally distributed random variables with singular distribution is normal if 2 and only if EI1]l 1< 00. Ya. R. Aead. Sei.

Such that for any (: > 0, We will give a necessary and sufficient condition for the stability of the means. We say that two systems l11Jnk 11 and 117Jnk 11 are equivalent if and Clearly for equivalent systems the means are simultaneously either stable or unstable. Theorem 8. A necessary and sufficient condition lorthe stability 01 the means U n, n ~ 1, is the existence 01 some system 117Jnkll equivalent to II1JnkIL lor which (13) Prool 01 the sufficiency. Theorem 1 implies that for any (: > 0, Therefore, (13) implies that and, from the equivalence of II1Jnk 11 and 117Jnk 11 we obtain (14) 24 ON SUMS OF INDEPENDENT RANDOM VARIABLES which proves the sufficiency.

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