# Tag Info

### Convergence and boundedness in probability for Op + op(1)

You have to prove that $a_N+b_N$ (it is better to keep indices) converges in probability to zero. Since the sum of two sequence that converge to $0$ in probability also converges in probability, it ...

### Strong consistency of kernel density estimator

First of all, you have not explained what the function $K_h(x)$ is. From a look at the book you referenced this is defined as $$K_h(x) = \frac{1}{h} K\left( \frac{x}{h} \right).$$ Therefore, we want ...
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### Switching integration and minimization for positive random variables?

No. Let $X$ be $1$ with probability $1/2$ and $0$ otherwise (i.e., a fair coin), $Y = 1-X$, and $Z = \min(X,Y) = 0$. Then $E(X) = E(Y) = 1/2$, but $E(Z) = 0$.
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### Switching integration and minimization for positive random variables?

Counter example: Let Z = min(X,Y), with sample (x, y) = (0,0) with probability 1/3, (0,3) with probability 1/3, and (3, 0) with probability 1/3. E[X] = E[Y] = 1. Z = 0, and E[Z] = 0. Any A between 0 ...

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