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How can I find the limiting distribution of $Z_n=\sqrt{n}\frac{X_1X_2+X_3X_4+\cdots+X_{2n-1}X_{2n}}{X_1^2+\cdots+X_{2n}^2}$?

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_i) = 0$, $\text{Var}(X_i) = 1$, and $E(X_i^4) < \infty$. We are tasked with finding the limiting distribution of $$ Z_n = \sqrt{n} \frac{...
Robert Long's user avatar
2 votes
Accepted

Do convergence rates for (convex) gradient descent apply when domain is (convex) subset of reals?

I think your assumption is mistaken. Most algorithms I worked with handle simple constrains on domains by automatically transforming to an unconstrained $\mathbb{R}^n$ under the hood. For your ...
Martin Modrák's user avatar
2 votes

non-positive-definite Hessian matrix/non-convergence problem with glmmTMB

Hard to say, exactly. If your response variable is positive with exact zeros then indeed using a zero-inflated Gamma or a Tweedie distribution as the response would make more sense and might work ...
Ben Bolker's user avatar
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2 votes

If the variance converges to zero, when do we have almost sure convergence

To show $\{X_n\}$ converges to $c$ almost surely, one can add the extra condition that $$\sum_{n=1}^\infty \mathrm{Var}(X_n)<\infty \tag{*}\label 1$$ which can be satisfied by, for example, if $\...
Mingzhou Liu's user avatar
3 votes
Accepted

Convergence of MLE for non-IID data

It will depend, but there are some things that can usefully be said, especially if this is a smooth parametric model (as seems to be implied) If you have a law of large numbers and central limit ...
Thomas Lumley's user avatar
4 votes
Accepted

If the variance converges to zero, when do we have almost sure convergence

No. Consider $X_n \in \{0,1\}$, therefore bounded, and the sequence of probabilities $P_n(X_n = 1) = 1/n$ (and therefore $P_n(X_n = 0) = 1 - 1/n$.) Clearly $X_n \to 0$ in probability, but Borel-...
jbowman's user avatar
  • 40.5k
3 votes

The sum of $O_p$ --$ O_p \left(s^2\frac{\log d}{n}+s\sqrt{\frac{\log d}{n}} \right) $

You can drop terms when you know they are smaller than some constant multiple of other terms. For example, because $s>1$, we know $$s^2\frac{\log d}{n}>s\frac{\log d}{n}$$ which is why the ...
Thomas Lumley's user avatar

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