query: Let $X_1,X_2,\cdots$ be i.i.d $p^{\text{th}}$ integrable random variables. Show that $\cfrac{1}{n^{1/p}} \sup_{k\le n} |X_k| \to 0$ almost surely.
positive_passages.docid: 70067
positive_passages.text: Asymptotics of $\max\limits_{1\leqslant k\leqslant n}X_k/n$
negative_passages.docid: 42366
negative_passages.text: Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
negative_passages.docid: 144923
negative_passages.text: Prove that the maximum of $n$ independent standard normal random variables, is asymptotically equivalent to $\sqrt{2\log n}$ almost surely.
negative_passages.docid: 47895
negative_passages.text: Prove $\lim\limits_{n\to\infty}n^{-1/(p+1)}(1^{1^p}\cdot 2^{2^p}\cdots n^{n^p})^{n^{-p-1}}=e^{-1/(p+1)^2}$
negative_passages.docid: 50084
negative_passages.text: Probability that a quadratic polynomial with random coefficients has real roots
negative_passages.docid: 155974
negative_passages.text: Show $||Tx||_2 \leq ||(v_n)||_2 \cdot ||x||_{\infty}$
negative_passages.docid: 7448
negative_passages.text: Prove $S \doteq \sum_{n=1}^{\infty} p_n  0$ assuming $0 \leq p_n < 1$.
negative_passages.docid: 56884
negative_passages.text: Non-decreasing sequence of random variable convergence in probability implies it also converges almost surely.
negative_passages.docid: 128599
negative_passages.text: Linearity of convergence in distribution of random variables
negative_passages.docid: 75030
negative_passages.text: Law of large numbers for non-identically distributed Bernoulli random variables
negative_passages.docid: 135663
negative_passages.text: Probability that a sequence of random variables converges to 0 or 1
negative_passages.docid: 155190
negative_passages.text: Stone-Weierstrass theorem with $p(1/x)$
negative_passages.docid: 14079
negative_passages.text: On the proof that every positive continuous random variable with the memoryless property is exponentially distributed
negative_passages.docid: 47320
negative_passages.text: An example of almost sure finite random time S such that $W_t=B_{S+t}-B_S$ is not a brownian motion
negative_passages.docid: 19167
negative_passages.text: Show that $X_{n}\sim N(\sin(\pi n),1)$ is uniformly tight
negative_passages.docid: 9498
negative_passages.text: If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN)
negative_passages.docid: 139877
negative_passages.text: Square of a uniform random variable