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May
5
accepted Show $\dbinom{n}{2}^{-1} \sum_{i < j} X_i X_j \xrightarrow{p} \mu^2$, when $X_i$ are i.i.d. with mean $\mu$ and finite variance
Apr
29
accepted $X_n \sim \text{Exponential}(\lambda_n)$, independent, $\sum 1/\lambda_n = \infty$, then, $\sum X_n=\infty$ a.s.
Apr
29
accepted Strict inequality in Reverse Fatou lemma: $\varlimsup \int f_n\le \int \varlimsup f_n$
Apr
23
accepted Show $\det \left[T\right]_\beta=-1$, for any basis $\beta$ when $Tx=x-2(x,u)u$, $u$ unit vector
Apr
22
accepted Is the set $A=\{1,2,\ldots,\omega\}$ internal, when $\omega$ is infinite
Apr
21
accepted Show $x\sqrt{n} - n \ln\left(1+\frac{x}{\sqrt{n}}\right) \to \frac{x^2}{2}$
Apr
20
accepted Does $f_n \to 0$, a.e., implies $\int_{\mathbb R} \sin(f_n(x)) dx \to 0$, when each $f_n \in L^1$
Apr
12
accepted If $f_n,g_n \in L^1$ and $f_n,g_n \to 0$, show $\int_A (2f_n g_n)/(1+f_n^2+g_n^2)\to 0$, when $A$ has finite measure
Apr
10
accepted If $L=\{B : BA = 0 \}$ and $R=\{C : AC = 0 \}$, what is the dimension of $L$, and $R$? $L,R,\{A\} \subset \mathbb R_{n \times n}$
Apr
10
accepted Find the common limit of $\frac{2}{1/a_n+1/b_n}$ and $\sqrt{a_n b_n}$
Apr
10
accepted If $\dim(A+B)=\dim(A\cap B)+1$, then $A \subset B$ or $B \subset A$
Apr
10
accepted Harmonic mean: show $\max\{ax,by\} \ge \frac{1}{a+b}(x+y)$, $a,b>1$, $x,y\ge 0$
Mar
9
accepted Compound Poisson process: distribution of stopped local time $L_T$
Feb
23
accepted $X_i \sim N(\theta,1), \theta \in \Bbb Z$: $T=\left\lfloor \bar X_n \right \rfloor$ not consistent for $\theta$
Feb
17
accepted Show the $l^2$ ellipsoid is close : $\left\{(x_n) \in l^2 : \sum\frac{x_n^2}{a_n^2} \le 1\right\}, a_n \to 0$
Feb
11
accepted $d\left(\left(x_1,x_2\right),\left(y_1,y_2\right)\right)=|x_1-x_2|+|x_1-y_1|+|y_1-y_2|$ : complete?
Feb
8
accepted $f$ continuous on a circle, then there is a diameter at which ends $f$ has the same value
Feb
7
accepted $T:\Bbb R^2 \to \Bbb R^2$, linear, diagonal with respect to any basis.
Feb
7
accepted $A$ skew-symmetric, then $x^T A^2 x \le 0$
Feb
2
accepted Pointwise limit of $f_n(0)=0$, $f_n(1/n)=n$, linear in between