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In Durrett's probability theory, Kolmogorov extension theorem is given as below:

Theorem 2.1.21 (Kolmogorov's extension theorem) Suppose we are given probability measure $\mu_n$ on $(\mathbf{R}^n, \mathcal{R}^n)$ that are consistent, that is, $$\mu_{n+1}((a_1, b_1] \times \cdots \times (a_n, b_n] \times \mathbf{R}) = \mu_n((a_1, b_1] \times \cdots \times (a_n, b_n])$$ then there is a unique probability measure $P$ on $(\mathbf{R}^\mathbf{N}, \mathcal{R}^\mathbf{N})$ with $$P(\omega: \omega_i \in (a_i, b_i], 1 \le i \le n) = \mu_n((a_1, b_1] \times \cdots \times (a_n, b_n])$$

where $\mathbf{N}$ stands for natural numbers.

My questions are:

(1) are all probability measures consistent (by the above definition)? I'm assuming they're not but in the above condition, we can write $\mu_{n+1}$ as $\mu_n((a_1, b_1] \times \cdots \times (a_n, b_n]) \mu_1(\mathbf{R}) = \mu_n((a_1, b_1] \times \cdots \times (a_n, b_n])$. This is always going to hold for any probability measure since $\mu_1(\mathbf{R}) = 1$ for any probability measure on a real line.

(2) What are some examples of non-consistent probability measure?

(3) the consistency condition is slightly different from the ones shown in Wikipedia where there are two consistency conditions, especially the permutation part:

For all $t_1, \cdots , t_k \in T$, $k \in \mathbb{N}$ let $\nu_{t_1,\cdots,t_k}$ be probability measures on $\mathbb{R}^{nk}$ s.t. $$\nu_{t_{\pi(1)},\cdots,t_{\pi(k)}} (F_1 \times \cdots \times F_k) = \nu_{t_1,\cdots,t_k} (F_{\pi(1)} \times \cdots \times F_{\pi(k)}) $$ for all permutations $\pi$ on $\{1,2, \cdots,k\}$ and $$\nu_{t_1,\cdots,t_k}(F_1\times \cdots \times F_k) = \nu_{t_1,\cdots,t_k, t_{k+1}, \cdots, t_{k+m}}(F_1\times \cdots \times F_k\times \mathbb{R}^n \times \cdots \times\mathbb{R}^n )$$ for all $m \in \mathbb{N}$

Is Durrett's condition of consistency weaker? How are they different?

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Let $f(x)$ be the PDF of a continuous random variable $X$.

Based on $f(x)$, we can build three different measures $\mu_n$, $\gamma_n$, $\beta_n$ to use as examples. For each positive integer $n$ and for each Borel measurable set $A \subseteq \mathbb{R}^n$ define

\begin{align} \mu_n(A) &= \int_{(x_1, ..., x_n) \in A} \left(\prod_{i=1}^n f(x_i)\right) dx_1...dx_n\\ \gamma_n(A) &=\int_{(x_1, ..., x_n) \in A} \left(\prod_{i=1}^n f(x_i-i)\right) dx_1...dx_n\\ \beta_n(A) &= \int_{(x_1, ..., x_n) \in A} \left(\prod_{i=1}^n f(x_i-n)\right) dx_1...dx_n\\ \end{align}

Then:

  • $\mu_n$ satisfies the consistency property of Durrett and treats all dimensions in the same way. It could be used to model i.i.d. random variables $\{X_i\}_{i=1}^{\infty}$.

  • $\gamma_n$ satisfies the consistency property of Durrett but treats different dimensions differently. It could be used to model mutually independent random variables $\{X_i\}_{i=1}^{\infty}$ where the variables have different distributions (where $X_i$ has PDF $f(x-i)$).

  • $\beta_n$ does not satisfy the consistency property of Durrett. It cannot be used to model random variables $\{X_i\}_{i=1}^{\infty}$. This is because the marginal distribution of $X_1$, when grouped with the first five variables $(X_1, ..., X_5)$, would be different from the marginal distribution of $X_1$ when grouped with the first six variables $(X_1, ..., X_6)$.

I observe that Durrett seems to be stating a theorem about probability measures on $\mathbb{R}^{\mathbb{N}}$. This can be viewed as a theorem about discrete time stochastic processes. The Wikipedia article seems to be stating a theorem in more generality, one which could apply to either continuous time or discrete time stochastic processes. So it uses more complex notation that is not needed when only discrete time stochastic processes are considered.


Note that the permutation notation in the Wikipedia description is similar to observing that if we have three different random variables $X_1, X_2, X_3$, we can write a CDF using any ordering we want: $$ F_{X_1, X_2, X_3}(x_1, x_2, x_3) = F_{X_2, X_3, X_1}(x_2, x_3, x_1) = P[X_1\leq x_1, X_2\leq x_2, X_3 \leq x_3]$$ It is useful to allow different orderings if we want to isolate $k+m$ points in time and group $k$ of them first and the remaining $m$ of them last, regardless of their actual ordering on the timeline.

This ordering issue is not required for discrete time because we would always list the dimensions in order. For example we would always write $F_{X_1, X_2, X_3}(x_1, x_2, x_3)$ and we would never write it as $F_{X_2, X_3, X_1}(x_2, x_3, x_1)$. So we could even simplify the indexing notation (as Durrett does) via:
$$ F_3(x_1, x_2, x_3) \equiv F_{X_1, X_2, X_3}(x_1, x_2, x_3)$$ since there is no ambiguity about ordering the first three random variables of the discrete time process $\{X_1, X_2, X_3, X_4, ...\}$.

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  • $\begingroup$ Thank you for the detailed answer! $\endgroup$
    – MoneyBall
    Jul 16 at 0:29

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