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2 votes

Writing a counting process $N(t)$ as $N(t) = \sum_{k=1}^{n(t)} Y_k$ : Existence and properties of $Y_k$?

See this paper [1] and references within: The usual approach is to bound the probability of $\{N(t) < n\}$ using an equivalent event related to $T(n) = \sum_{i = 1}^...
Ziv's user avatar
  • 336
1 vote

Limit of expectation of ratio of sums of uniformly distributed variables

By the strong law of large numbers,$\frac{U_1^a+\cdots+U_n^a}{U_1^b+\cdots+U_n^b}$ converges almost surely to $\frac{b+1}{a+1}$ as $n$ goes to infinity. Moreover it is positive and bounded by $1$ (...
Will's user avatar
  • 7,337
1 vote

Rate of convergence (Berry-Esseen theorem) for the sum of an asymptotically normal random number of random variables?

I learned that there is older literature on the sum of a random number of random variables that can be put to work ... and that there are opportunities for going further. The following is from Korolev ...
PtH's user avatar
  • 1,144
1 vote

Is central limit theorem applicable to Poisson distributed samples multipled with different coefficients

I really don't understand the physical details of the scenario and how that would translate into a probability model of the measurements. But, if your question is about whether a sum of independent, ...
Guillaume Dehaene's user avatar
1 vote

Finding Probability Limit

If you want a direct argument, you can just modify the proof of the WLLN (for square integrable independent RVs), which is short anyways. Assuming your samples are i.i.d., then $E[W_n] = \frac{n-1}{n}\...
daisies's user avatar
  • 1,628
1 vote

Approximating a discrete distribution with CLT

If iid $Y_i\sim \text{Poisson}(\lambda)$ for $i=1,2,\dots,n$ and denote their average as $\bar{Y}$, then their sum $$ n\bar{Y}\sim \text{Poisson}(n\lambda) $$ exactly without approximation. Now if ...
Zack Fisher's user avatar

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