# Bernoulli experiment

Let $$\lambda = \frac{m}{n}$$ where $$m and $$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!!

• What do mean by $k_{i}\rightarrow i \cdot m$? Commented Feb 5, 2020 at 10:32
• 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. Commented Feb 5, 2020 at 10:35
• Are the variables $Y_i$ independent? Commented Feb 5, 2020 at 11:12
• Thanks for the question, $Y_i$'s are not independent. I will edit my question. Commented Feb 5, 2020 at 11:30