Disclaimer: This article does not constitute investment advice.

In an era marked by severe monetary expansion, intense geopolitical tensions, global instability, and the steady depletion of natural resources, where resources are increasingly becoming the ultimate source of value, gold is destined to rise substantially over the long term. The magnitude of that rise may well exceed anything people can judge from past experience. That broad trend is unlikely to change. In the short run, however, factors such as short squeezes triggered by higher futures margin requirements and deliberate market intervention by large capital can create enormous volatility and significant near-term risk. In short, gold remains a long-term bullish asset, but it also carries considerable short-term risk. We should therefore adhere to the basic principle of keeping positions light, investing for the long term, and investing with peace of mind.

Introduction

Markov Chain

Let $(\Omega,\mathcal{F})$ be a measurable space, $P:\Omega\times \mathcal{F}\to [0,1]$ be a transition kernel, that is,

• for every $x\in \Omega$, $P(x,\cdot)$ is a probability measure on $\mathcal{F}$;
• for every $A\in \mathcal{F}$, $x\mapsto P(x,A)$ is measurable.

This is the talk I presented in the group seminar, titled: Sampling and Diffusion Model.

This is the talk I presented in the group seminar, titled: Analogy between Sampling and Optimization.

In revisiting the proofs of many theorems in mathematics, we often need to use sequential compactness. However, the Riesz lemma in normed linear spaces tells us that every bounded set in an infinite-dimensional normed linear space is not sequentially compact, which brings many inconveniences. In fact, the essential reason for this phenomenon is that the topology induced by the norm in a normed linear space is “too strong.” For this reason, we need to introduce the weak topology, so that bounded sets are sequentially compact in the weak-topology sense. In this article, we mainly review local convex topological vector spaces and some main results on weak convergence.

The Wasserstein distance has wide applications in problems such as optimal transport. It characterizes the distance between two probability measures. In German, Wasser means water, and Stein means stone (although I have not carefully verified whether Wasserstein is a person’s name; after all, many German names are of this kind, for example: Einstein = one stone). This article will mainly answer the following two questions:

• On what set is the Wasserstein distance defined?
• Why is the Wasserstein distance a distance? That is, why does it satisfy the three axioms of a metric?

In this article, we only consider the Wasserstein distance on the $d$-dimensional Euclidean space $\mathbb{R}^d$. Since $\mathbb{R}^d$ is a special Polish space, the conclusions of this article can also be extended to general Polish spaces.

How can one define a product probability measure on infinitely many probability spaces? That is, given a family of probability spaces $(\Omega_t,\mathcal F_t)$, $t\in T$, suppose that for every nonempty finite subset $S\subset T$, a probability measure $\mathbb P_S$ has already been defined on $\prod_{t\in S}\mathcal F_t$. Then how can we define a “suitable” probability measure $\mathbb P$ on the product measurable space

$$
\left(\prod_{t\in T}\Omega_t,\prod_{t\in T}\mathcal F_t\right)
$$

such that its marginal distribution measures are exactly $\mathbb P_S$?

Unless otherwise specified, all classes of sets in this article refer to classes of subsets of $\Omega$.

The Monotone Class Theorem

Stochastic Processes and Paths

Definition 1: Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space, and let $(X_t)_{t\in T}$ be a family of random variables on $(\Omega,\mathscr{F},\mathbb{P})$, that is, for every $t\in T$, we have
$$
X_t: (\Omega,\mathscr{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))
$$
measurable. Then
$$
\begin{aligned}
X:T\times\Omega&\to \mathbb{R}\\\
(t,\omega)&\mapsto X_t(\omega)
\end{aligned}
$$
is called a stochastic process. For fixed $\omega\in \Omega$, we call
$$
\begin{aligned}
X_\omega:T&\to \mathbb{R}\\\
t&\mapsto X(t,\omega)
\end{aligned}
$$
a path, and call the collection of all paths
$$
\mathbb{D} := \{X_\omega\ ;\ \omega\in \Omega\}
$$
the path space.

Remark: Each element $X_\omega$ of the path space is a mapping from $T$ to $\mathbb{R}$.

The KKT conditions (Karush–Kuhn–Tucker Conditions) are one of the most important results in optimization, and readers familiar with optimization will certainly not find them unfamiliar. This article will focus on introducing the motivation behind some key concepts introduced in the proof of the KKT conditions (such as the introduction of the active set and the constraint qualification), and will review the complete proof of the KKT conditions.