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Let $\lambda = \frac{m}{n}$ where $m<n$ and $m, n$ are positive integers. Let $\{k_i\}$ be a sequence of non-negative reals. For $i=1,2,3,\dots,$ let $X_i$ be a binomial random variable with parameters $n$ and $\lambda$ and $t_i = i\cdot m +k_i.$ Let $Y_i = X_1+X_2+\dots X_i$ for $i = 1,2,3,\dots .$ Is it possible to choose a decreasing sequence of non-negative reals $\{k_i\}$ such that $k_i \rightarrow 0$ as $i \rightarrow \infty$ and $P(\cap_{i=1}^{\infty}\{Y_i \leq t_i\})$ is close to 1 ?

I will try to explain the physical problem which am trying to model so that the above formulation may make sense. There is a bernoulli random variable $X$ with probability of failure $\lambda.$ After $i \cdot n$ number of trials, if the process was truly a bernouli random variable with parameter $\lambda$, I will observe a total number of failures which is close to $i\cdot m.$ After first $n$ trials, I examine the total number of failures. If the total number of failures is much greater than $m,$ then I suspect that the process is not a Bernouli with parameter $\lambda$ and stop the process. (I stop the process only if the process is Bernoulli with parameter strictly greater than $\lambda$ i.e., I can afford a failure rate of atmost $\lambda$). Otherwise I continue with the next $n$ trials and so on. So each $i$th block has $n$ independent trials of the bernoulli r.v $X,$ given that the process did not stop till the first $i-1^{th}$ blocks. The problem then is in some sense to find the correct sequence of thresholds $k_i$ after each $i^{th}$ block, so that I stop the process only when I have strong evidence that the process is a Bernoulli with parameter greater than $\lambda.$

Any help with the problem will be greatly appreciated!!

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  • $\begingroup$ What do mean by $k_{i}\rightarrow i \cdot m$? $\endgroup$ Commented Feb 5, 2020 at 10:32
  • $\begingroup$ Intuitively I think that for large $i,$ the threshold $k_i$ should be close to $i\cdot m.$ But may be the notation $k_i \rightarrow i \cdot m$ is incorrect. $\endgroup$
    – egt123
    Commented Feb 5, 2020 at 10:35
  • $\begingroup$ Are the variables $Y_i$ independent? $\endgroup$
    – leonbloy
    Commented Feb 5, 2020 at 11:12
  • $\begingroup$ Thanks for the question, $Y_i$'s are not independent. I will edit my question. $\endgroup$
    – egt123
    Commented Feb 5, 2020 at 11:30

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