Why Finite-Dimensional Laws Must Agree
A finite-dimensional law describes the joint distribution of finitely many coordinates. But a proposed collection of such laws cannot be chosen independently: the law for several coordinates must give the stated law for fewer coordinates when the omitted coordinates are forgotten. The order used to list coordinates also matters in a precise way: reordering the list must reorder the coordinates of the law, not change their values or their labels.
We formulate these requirements for a family of measurable spaces \((S_i,\mathcal{S}_i)\), indexed by a set \(T\). For a finite ordered list \(\mathbf{i}=(i_1,\ldots,i_n)\) of distinct indices, write \(S_{\mathbf{i}}=\prod_{r=1}^n S_{i_r}\), equipped with the product sigma-algebra \(\bigotimes_{r=1}^n\mathcal{S}_{i_r}\). A law \(P_{\mathbf{i}}\) is a probability measure on that measurable space. For the empty list, take the product space to be a singleton and its law to assign mass one to that point.
More explicitly, let \(\pi_k:S_{\mathbf{i}}\to S_{(i_1,\ldots,\widehat{i_k},\ldots,i_n)}\) be the projection that drops coordinate \(k\); the hat means that this entry is omitted. For a measurable map \(F\), the pushforward \(F_{\#}P\) is the measure defined by \(F_{\#}P(A)=P(F^{-1}(A))\). Deletion says
If \(\sigma\) is a permutation of \(\{1,\ldots,n\}\), define \(T_\sigma(x_1,\ldots,x_n)=(x_{\sigma(1)},\ldots,x_{\sigma(n)})\), with target space \(S_{(i_{\sigma(1)},\ldots,i_{\sigma(n)})}\). Permutation consistency says
These identities do not say that the laws are invariant under permutations. Invariance would assert equality with the original law after reordering, which is an additional property and need not hold. Consistency instead says that the law is transformed along with the coordinate labels.
Why Joint Laws Satisfy the Identities
Suppose \((\Omega,\mathcal{F},\mathbb{P})\) is a probability space and \(X_i:\Omega\to S_i\) are measurable random variables. The joint law \(P_{\mathbf{i}}\) is the distribution of the random vector \((X_{i_1},\ldots,X_{i_n})\): for a measurable set \(A\subseteq S_{\mathbf{i}}\), it assigns probability \(\mathbb{P}((X_{i_1},\ldots,X_{i_n})\in A)\).
Proof. Fix a list \(\mathbf{i}\) and a coordinate position \(k\). Applying \(\pi_k\) to the random vector \((X_{i_1},\ldots,X_{i_n})\) gives exactly the vector with \(X_{i_k}\) removed. Thus, for any measurable set \(A\) in the shortened product space, the event that the shortened vector lies in \(A\) is the event that the original vector lies in \(\pi_k^{-1}(A)\). The two probabilities are equal, which is the deletion identity. For a permutation \(\sigma\), applying \(T_\sigma\) to the original vector gives \((X_{i_{\sigma(1)}},\ldots,X_{i_{\sigma(n)}})\). For any measurable set in this reordered product space, the same preimage argument gives the permutation identity. These arguments include the empty list, whose random vector is the constant singleton-valued vector and whose law has mass one. \(\square\)
This proposition gives a necessary condition for a family to be the finite-dimensional laws of a process. At this stage, consistency is a set of requirements on the proposed finite laws; it does not itself construct a process or an infinite-dimensional probability measure.
Checking the Identities in Finite-State Models
Worked Example: A Consistent Pair of Binary Laws
Take two coordinates with state space \(\{0,1\}\), equipped with all its subsets. Rows in the table below record the first coordinate and columns the second. Define the pair law and the two one-coordinate laws as follows:
All pair entries are nonnegative, and their sum is \(1/2+1/6+1/6+1/6=1\), so the table defines a probability law. Deleting the second coordinate means summing each row: the first row gives \(1/2+1/6=2/3\), and the second gives \(1/6+1/6=1/3\). These are exactly \(P_{(1)}(0)\) and \(P_{(1)}(1)\). Deleting the first coordinate means summing each column: the first gives \(1/2+1/6=2/3\), and the second gives \(1/6+1/6=1/3\). These are exactly \(P_{(2)}(0)\) and \(P_{(2)}(1)\).
For the swap, the entry at \((a,b)\) in the reordered law must equal the original entry at \((b,a)\). The table is symmetric: its off-diagonal entries are both \(1/6\), and its diagonal entries remain unchanged. Thus the pair law also satisfies permutation consistency. As a normalization check, each one-coordinate law has total mass \(2/3+1/3=1\).
Worked Example: A Proposed Marginal That Fails
Let the proposed pair law on \(\{0,1\}^2\) assign mass \(1/4\) to each of its four points. Suppose the proposed first-coordinate law instead assigns mass \(3/4\) to \(0\) and \(1/4\) to \(1\). The pair table is
Deleting the second coordinate requires summing across each row. The mass assigned by the pair law to \(X_1=0\) is \(1/4+1/4=1/2\), not \(3/4\). The mass assigned to \(X_1=1\) is also \(1/4+1/4=1/2\), not \(1/4\). Therefore the proposed pair and singleton laws are inconsistent. Both are probability laws individually—the four pair entries sum to \(1\), and \(3/4+1/4=1\)—but separate normalization does not guarantee consistency.
Check: the actual first-coordinate marginal has total mass \(1/2+1/2=1\). Its failure is not a normalization error; it is a disagreement with the separately proposed singleton law.
Worked Example: A Permutation-Symmetric Triple
Again take state space \(\{0,1\}\) at each of three coordinates. Assign mass \(1/4\) to \(000\), mass \(1/12\) to each of \(001,010,100\), mass \(1/12\) to each of \(011,101,110\), and mass \(1/4\) to \(111\). The total mass is \(1/4+3(1/12)+3(1/12)+1/4=1\). Every point has nonnegative mass, so this defines a probability law.
The mass depends only on the number of coordinates equal to \(1\), so every permutation of the coordinates leaves the point masses unchanged. This verifies permutation consistency: after reordering, each particular triple still has the same probability. To check deletion, sum over the third coordinate. The resulting law on the first two coordinates is
The entries sum to \(1/3+1/6+1/6+1/3=1\). Summing rows gives first-coordinate probabilities \(1/3+1/6=1/2\) and \(1/6+1/3=1/2\). Summing columns gives the same two probabilities for the second coordinate. Permutation symmetry gives these same pair and singleton marginals whichever coordinates are retained. Thus the triple law, its pair marginals, and its singleton marginals agree under both deletion and reordering.
Check: the total mass of the triple law and of each displayed marginal is \(1\), and each marginal entry was obtained by summing exactly the triple outcomes projecting to that entry.
Elementary Identities Are Enough
For a long list, checking every deletion position and every permutation may seem like many separate requirements. It is enough to check deletion of the last coordinate and swaps of adjacent coordinates. The reason is that any coordinate can be moved to the end by a finite sequence of adjacent swaps, and any reordering can be obtained by such swaps.
Proof. If the family satisfies the full consistency definition, then deletion of the last coordinate is one of its deletion identities, and an adjacent swap is one of its permutation identities. This proves necessity.
Conversely, assume the two stated elementary conditions. Any permutation of a finite list can be written as a finite composition of adjacent swaps: for example, move the entry required in the first position leftward one place at a time, then do the same for the second position among the remaining entries, and continue. Each adjacent-swap identity pushes the law forward under the corresponding coordinate-reordering map. Pushforwards compose: for measurable maps \(F,G\), \((G\circ F)_{\#}P=G_{\#}(F_{\#}P)\), since both sides assign to a measurable set \(A\) the value \(P(F^{-1}(G^{-1}(A)))\). Applying this fact successively gives permutation consistency for the entire composition.
Now fix a list and a coordinate position \(k\) to delete. Use adjacent swaps to move that coordinate to the last position, keeping track of the coordinate labels and the resulting reordered list. By the permutation identity just proved, the law of that reordered list is the pushforward of the original law under the reordering map. Delete its last coordinate using the assumed last-coordinate identity. The resulting vector consists, in order, of the original coordinates other than \(k\). The composite of the reordering map and this last-coordinate projection is exactly the projection \(\pi_k\), followed by the identity ordering on the retained coordinates. The composition rule for pushforwards therefore gives \((\pi_k)_{\#}P_{\mathbf{i}}=P_{(i_1,\ldots,\widehat{i_k},\ldots,i_n)}\). When the list has length one, the last-coordinate identity says its projection has the unit law on the empty product, as required. Thus all deletion identities hold, and the family is consistent. \(\square\)
A Common Permutation Error
For a nonsymmetric pair law, swapping coordinates generally changes the table. Consider the law with \(P(0,0)=1/2\), \(P(0,1)=1/4\), \(P(1,0)=0\), and \(P(1,1)=1/4\). Its total mass is \(1/2+1/4+0+1/4=1\). The law after swapping the coordinates must put the original mass at \((b,a)\) into the new entry at \((a,b)\). Its table is therefore
A common incorrect calculation is to copy the original table unchanged as the swapped law. That would make the probability that the first coordinate is \(0\) equal to \(1/2+1/4=3/4\). But the swapped law must have as its first-coordinate marginal the original second-coordinate marginal, whose probability of \(0\) is \(1/2+0=1/2\). In the correctly transposed table, the first row sums to \(1/2+0=1/2\), as required. The correction is to move each point \((a,b)\) to \((b,a)\), not to assume that the law itself is unchanged.
Consistency is the basic compatibility check on finite-dimensional probability laws: every smaller law must be the appropriate marginal of every larger one, and every reordered list must carry the correspondingly reordered law. These conditions prevent contradictions between descriptions of overlapping finite collections of coordinates. They are necessary for a process to have the proposed finite-dimensional distributions; by themselves, they are not an existence theorem for such a process.
Check Your Understanding
For each question, identify the relevant coordinate map and check the probability masses explicitly where appropriate.
- For a law on three coordinates, write the deletion identity for removing the second coordinate using pushforward notation.
- A pair table on \(\{0,1\}^2\) has rows \((1/3,1/6)\) and \((1/6,1/3)\). What singleton law does deletion require for either coordinate?
- Explain why an unchanged table after swapping coordinates is valid for some laws but not for every law.
- Why does checking deletion of the last coordinate alone not establish full consistency for ordered lists?
- State why consistency of finite-dimensional laws is necessary for a process, and why it does not by itself construct one.