Abstract
Let $X_1,X_2,\ldots$ be a sequence of independent replicates of a random variable $X$ and let $\{N_t\}_{t\geq 0}$ be a non-negative integer valued random process and assume that $\{N_t\}_{t\geq 0}$ and $X$ are independent. Then, under some conditions it is shown that the probability $P(\sum_{i=1}^{N_t} X_i\geq 0)$ decays exponentially fast as $t\to\infty$. Moreover, we consider a testing problem in a compound Poisson process, and we study the exact slope of a test statistic based on the sum of random number of independent and exponentially distributed random variables.
Citation
Haruyoshi MITA. "Probabilities of Large Deviations for Sums of Random Number of I.I.D. Random Variables and Its Application to a Compound Poisson Process." Tokyo J. Math. 20 (2) 353 - 364, December 1997. https://doi.org/10.3836/tjm/1270042109
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