= Solution
Write $\lambda=\sum_i\lambda_i$, and suppose first that $\lambda>0$. Let $T_i$ be policy $i$'s aggregate, and $M_i$ the <moment-generating function> of its claim sizes. The <Poisson distribution> count has <probability generating function> $G_i(z)=\exp(\lambda_i(z-1))$, so the <random-sum transform identity> and <independence> of the policies give
$$
M_T(u)=\prod_i\exp(\lambda_i(M_i(u)-1))
=\exp\left[\lambda\left(\sum_i\frac{\lambda_i}{\lambda}M_i(u)-1\right)\right].
$$
The expression in the weighted sum is the <moment-generating function> of a <mixture distribution> with claim <cumulative distribution function>
$$
F_*(x)=\sum_i\frac{\lambda_i}{\lambda}F_i(x).
$$
Thus \b[the aggregate has a <compound Poisson distribution> with intensity $\lambda$ and severity law $F_*$]. This identification remains valid without positive <exponential moments>, using the same calculation for <Laplace transforms of nonnegative random variables> and their uniqueness. It is the <Poisson superposition of insurance portfolios>.
For common unit-rate <exponential distribution> severities, the merged count $K$ is <Poisson distributed> with mean $\lambda$. Because each severity is strictly positive, $T=0$ exactly when $K=0$, giving
$$
\boxed{a=e^{-\lambda}.}
$$
For $k\ge1$, the sum of $k$ unit-rate <exponential distribution> claims has <Erlang distribution> density $x^{k-1}e^{-x}/(k-1)!$. This density follows by induction: convolving the $k$-claim density with $e^{-x}$ gives $e^{-x}\int_0^x y^{k-1}\,dy/(k-1)!=e^{-x}x^k/k!$. Conditional on a positive aggregate, the count weights are
$$
\mathbb P(K=k\mid K>0)=\frac{e^{-\lambda}\lambda^k/k!}{1-e^{-\lambda}}=\frac{\lambda^k}{(e^\lambda-1)k!}.
$$
Therefore the <hurdle decomposition of a positive random sum> has positive-component <probability density function>
$$
\boxed{\widetilde f_T(x)=\sum_{k=1}^{\infty}\frac{\lambda^k}{(e^\lambda-1)k!}\frac{x^{k-1}e^{-x}}{(k-1)!},\qquad x>0.}
$$
Each density integrates to one and the weights sum to one. Hence $F_T(x)=a+(1-a)\widetilde F_T(x)$ for $x\ge0$, where $\widetilde F_T$ is the conditional <cumulative distribution function> given $T>0$. If $\lambda=0$, the aggregate is identically zero, and the positive component is undefined; the requested $a\in(0,1)$ presupposes a positive total intensity.
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