6. Building Mult-Decrement Tables from Associated Single Life Functions

To understand the relationship between multi-decrement and the associated single decrement tables, first note that
begin{eqnarray*}
~ _t p_x^j =exp left{ – int_0^tmu_{x+u}^{0j} ~ du right}
end{eqnarray*}
and so (0 le ~ _t p_x^j le 1). Thus
begin{eqnarray*}
~ _s p_x^{00} = prod_{k=1}^n ~ _s p_x^k leq ~
_s p_x^j ,
end{eqnarray*}
for any (j). Thus, we have
begin{eqnarray*}
~ _t p_x^{0j} = int_0^t ~ _s p_x^{00} mu_{x+s}^{0j} ~ ds leq
int_0^t ~ _s p_x^j mu_{x+s}^{0j} ~ ds = ~ _t q_x^j .
end{eqnarray*}

That is, the multi-decrement transition probability is less than or equal to the associated single decrement function.

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