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You don't want a proof showing the MLE is never consistent with an infinite parameter space, because that's not true. There are many settings with countably infinite parameter spaces that have consistent MLEs. There are even many settings with uncountable parameter spaces that have consistent MLEs -- the usual $N(\mu,\sigma^2)$ model with real $\mu$ and ...
Goal: show $P(Y_n \le t) \to P(X \le t)$ [for $t$ at which $F_X(t):=P(X \le t)$ is continuous]. Hints: $P(Y_n \le t) = P(Y_n \le t, |X_n - Y_n| > \epsilon) + P(Y_n \le t, |X_n - Y_n| \le \epsilon)$. $P(Y_n \le t, |X_n - Y_n| > \epsilon) \le P(|X_n - Y_n| > \epsilon)$. What does the right-hand side converge to? $P(Y_n \le t, |X_n - Y_n| \le \epsilon)... 1 The first thing to try is always Chebyshev's inequality -- especially with well-behaved distributions where moments are likely to be enough to decide the question. $$\mathrm{var}[S_p]=\frac{1}{p^2}\sum_{i,j=1}^p \mathrm{cov}[a_iU_i^2,a_jU_j^2]$$ Now, correlations are bounded by 1,so: $$\left|\mathrm{cov}[a_iU_i^2,a_jU_j^2]\right|\leq \sqrt{\mathrm{var}[... 1 The difficulty disappears when you are careful in formulating the limits. In the first case, p is not constant, so it would be more precise to write it as p_n, as p varies with n. We can write n \cdot p_n \to \lambda>0 another way as p_n \sim \lambda/n, where \sim means that the quotient between the two sides converges to unity with n \to\... 1 Here is my attempt to explain the similarities and differences between the CLT and consistency from a statistical point of view using a particular example. My main focus here is intuition and I completely understand that this example is not flawless but I hope this is still useful. Suppose that X_1,\ldots,X_n are iid random variables such that \... 16 Consistency is a property of an estimator. The central limit theorem is, well, a theorem: it relates to the asymptotic property of the sample average under certain conditions, and that they tend to a normal distribution with variance equal to the inverse of the information matrix at a rate of root-n. Not all estimators are sample averages. And if they're ... 1 Note that this expectation does not exist for all g, because if g is sufficiently fast-growing then E[g(\bar X_n)] may not be finite for any n. For example, this happens if each X_i follows a standard normal distribution, so that \bar X_n \sim N(0, \frac 1 {\sqrt{n}}), and g(x) = e^{x^8}. (Though the specific g that you are most interested in ... 2 Let us begin with the example g(x)=|x| 1_{|x|>c}. Suppose that X_1,\dots, X_n are centered and have finite second moment \sigma^2. Denote F_n(x)=\mathbb{P}\left(\frac{1}{\sigma\sqrt{n}}\overline{X}_n\le x\right). We work with \frac{1}{\sigma \sqrt{n}}\overline{X}_n instead of \overline{X}_n as it simplifies the computations, please change c ... 0 Perhaps I could answer the part about your intuition on convex functions. For a convex and differentiable function f:\mathbb{R}^{n} \to \mathbb{R} which has a L-smooth gradient ||\nabla f(x) - \nabla f(y)|| \leq L||x-y||_{p}, (standard) gradient is guaranteed to converge assuming a fixed step size t \leq \frac{1}{K}. In this case, the convergence rate ... 0 Start by showing that X=O_p(1), then you can take advantage of the distribution of X_n being close to that of X to show that for large n, X_n can't be much more likely than X to exceed any specified bound M 0 The statement the authors intended is (A) for the following reasons. Addressing the confusion. Much of your reluctance to consider (A) as the correct statement amounted to insufficient attentiveness concerning the following basic distinction between:$$\mathbb{P} \left( \sup_{f \in \mathcal{F}} \lvert \hat{R}_n(f) - R(f) \rvert \geq \epsilon \right) = \... 0 When I started reading this question I was a bit confused. This factor$\sqrt{n\log n}\$ is not intuitive to me. It is not the typical expression in the CLT. Below I am trying to view your question in a more intuitive way without resorting to characteristic functions and looking at the limits of higher moments (which would be something mimicking the proof of ...