## 1 Introduction
### Background  
A Markov chain is a sequence of random variables \(X_{1},X_{2},X_{3},\ldots\) that satisfies the Markov property, which states that the probability of the system transitioning to the next state depends solely on the current state of the system and not the preceding events/states. The Markov property can be mathematically expressed as follows:  
\[P(X_{n+1}=x\mid X_{0}=x_{0},X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n})=P(X_{n +1}=x\mid X_{n}=x_{n}) \tag{1}\]  
In this equation, \(P\) denotes a probability distribution for the state of the system. \(X_{i}\) is a vector-valued variable representing the state at the \(i\)th step, and \(x,x_{0},x_{1},x_{2},\ldots,x_{n}\) are the specific state vector values of the system. Given a data point \(x_{0}\) from the actual data distribution, we can define a Markov chain--referred to as a forward diffusion process--by iteratively adding a multivariate Gaussian noise vector to \(x_{0}\). More specifically, at state \(X_{t-1}\), where the system has a specific state of \(x_{t-1}\), we add Gaussian noise--characterized by a mean vector of \(\mu_{t}\) and a covariance matrix of \(\Sigma_{t}^{2}\)--to \(x_{t-1}\) to transition to a state \(X_{t}\) with a specific value of \(x_{t}\). The probability distribution for this transition can be described as \(q(X_{t}=x_{t}\mid X_{t-1}=x_{t-1})=\mathcal{N}(x_{t};\mu_{t}=\sqrt{1-\beta_{t} }x_{t-1},\;\Sigma_{t}^{2}=\beta_{t}I)\), where \(\beta_{t}\) is a scalar; \(I\) is an \(m\times m\) identity matrix; and \(m\) is the cardinality of \(x_{i}\;(0\leq i\leq T)\). Considering the Markov chain property, we have  
\[q(x_{1:T}|x_{0})=\prod_{t=1}^{T}q(x_{t}\mid x_{t-1})\]  
where \(q(x_{1:T}|x_{0})\) is a short notation for \(q(x_{1},x_{2},\ldots,x_{T}|x_{0})\). A state \(x_{t}\) can be achieved by iteratively applying the Gaussian noise to \(x_{0}\)\(t\) times. Instead of an iterative process, this could also be achieved in a single step, utilizing a reparametrization trick, in which \(x_{t}\) is rewritten as  
\[x_{t}=\sqrt{1-\beta_{t}}x_{t-1}+\sqrt{\beta_{t}}\;\epsilon \tag{2}\]  
where \(\epsilon\) is an \(n\)-dimensional vector with standard normal distribution \(\mathcal{N}(0,I)\). It can be inferred that this reparametrization preserves the distribution of \(x_{t}\), i.e., \(x_{t}\sim\mathcal{N}(\mu_{t}=\sqrt{1-\beta_{t}}x_{t-1},\;\Sigma_{t}^{2}=\beta _{t}I)\). By changing the notation to \(\alpha_{t}=1-\beta_{t}\), Equation (2) can be written as:  
\[x_{t}=\sqrt{\alpha_{t}}x_{t-1}+\sqrt{1-\alpha_{t}}\;\epsilon \tag{3}\]  
As Equation (3) is a recursive equation, we can further expand the equation. \[x_{t}= \sqrt{\alpha_{t}}x_{t-1}+\sqrt{1-\alpha_{t}}\;\epsilon\] \[= \sqrt{\alpha_{t}}\Big{(}\sqrt{\alpha_{t-1}}x_{t-2}+\sqrt{1- \alpha_{t-1}}\;\epsilon\Big{)}+\sqrt{1-\alpha_{t}}\;\epsilon\] \[= \sqrt{\alpha_{t}\alpha_{t-1}}x_{t-2}+\sqrt{\alpha_{t}(1-\alpha_ {t-1})}\;\epsilon+\sqrt{1-\alpha_{t}}\;\epsilon \tag{4}\]  
Since \(\epsilon\) has a standard normal distribution \(\mathcal{N}(\mu=0,\;\Sigma^{2}=I)\), \(\sqrt{\alpha_{t}(1-\alpha_{t-1})}\;\epsilon\) has a normal distribution \(\mathcal{N}(\mu=0,\;\Sigma^{2}=\alpha_{t}(1-\alpha_{t-1}))\) and \(\sqrt{1-\alpha_{t}}\;\epsilon\) has a normal distribution \(\mathcal{N}(\mu=0,\;\Sigma^{2}=1-\alpha_{t})\). Consequently, their summation, i.e., \(\sqrt{\alpha_{t}(1-\alpha_{t-1})}\epsilon+\sqrt{1-\alpha_{t}}\epsilon\)has a normal distribution \(\mathcal{N}(\mu=0,\ \Sigma^{2}=1-\alpha_{t}\alpha_{t-1})\).