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Show that a.s. $\limsup_n \frac{X_n}{\sqrt{2\log n}} \le 1$

How to verify that an $\mathbb F[G]$-module has an imprimitivity decomposition or not?

Trivial representation from the Young symmetrizer

A question about $G$-modules, representations, and $F[G]$-modules

Probability of convergence of random variables

Let $\{X_n\}$ be i.i.d $N(0,1)$ random variables. Show that $\limsup_{n\rightarrow\infty} \frac{|X_n|}{\sqrt{\log n}}=\sqrt2$ a.s.

For non-negative definite symmetric matrices, $\mathrm{tr}(AB)\le \mathrm{tr}(A)\mathrm{tr}(B)$ [duplicate]

Orthogonal derivative implies second derivative is null

Does $\lim_{n} \langle x^{},f_n(t)\rangle =\langle x^{},f(t) \rangle$ for all $x^\in X^$ and a.e. $t$ imply $\lim_nf_n (t) = f (t)$ for a.e. $t$?

What is the intuition behind the law of quadratic reciprocity?

problem 6.7 in Character theory of finite groups

Show that $\forall f\in C(X)\ \ f(x_n) \to f(x)\implies x_n \to x$.

$\mbox{tr}((AB)^2) = \mbox{tr} (A^2B^2)$

Prove that $\det(AB - BA) = \frac{1}{3}\left(\mathrm{Trace}(AB - BA)^3\right)$

Why these are isomorphic each other for given these rings?

Show $\mathbb Z[x]/(x^2-cx) \ncong \mathbb Z \times \mathbb Z$.

Proving that a group of order $p^nq$ for primes $p$ & $q$ is not simple.

Fraction Field and isomorphism of R-modules

Show that for every $g \in G$, $g$ lies in the subgroup generated by $P$ and $g^{−1}Pg$.

Lower Semicontinuous Function = Supremum of Sequence of Continuous Functions

Let $S\subseteq\Bbb N $ and $X,Y\sim\text{Uniform}([0,1])$. What is $\text{Pr}(\lfloor X/Y\rfloor\in S)$?

Lebesgue measure is ergodic on a circle with irrational rotation.

Normal subgroup fixes every element of a set.

Prove this exercise from K.L. Chung using just Borel Cantelli Lemma

Show that if $\lim_{n\rightarrow\infty}\int_E|f_n|=\int_E|f|$ then $\int_E|f-f_n|\rightarrow 0$. [duplicate]

Change of Measure under Radon Nikodym Derivatives

How to prove that conditional independence does not imply independence?

Exercise 6.11 from Isaac's Character theory of finite groups.

How many binary strings of length $n$ contain $k$ flips?

Understanding the difference between the multinomial coefficients and the Stirling number of the 2nd kind