Finite-Dimensional Distributions

From a Process to a Finite Random Vector

A stochastic process is a collection of random variables indexed by time or another index set. Its values at any finite selection of indices form an ordinary random vector. The probability law of that vector is called a finite-dimensional distribution. It records how the selected values occur together, not just how often each value occurs separately. This makes it a basic way to describe the joint probabilistic behaviour of a process without needing to specify an entire sample path.

Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, let \(T\) be an index set, and let \(S\) be a measurable state space. A process \((X_t)_{t\in T}\) consists of measurable random variables \(X_t:\Omega\to S\). For distinct indices \(t_1,\ldots,t_k\), the vector \((X_{t_1},\ldots,X_{t_k})\) takes values in \(S^k\). Its probability law assigns to each measurable set \(B\subseteq S^k\) the probability that the vector lies in \(B\).

The indices are ordered because the coordinates in the vector follow that order. One may list them increasingly when \(T\) is an ordered time set, but the definition also works for any chosen order. Reordering the indices reorders the coordinates. Repeated indices add no new information: for example, \((X_s,X_s)\) is supported only on pairs whose two coordinates are equal. It is usually simpler to use distinct indices.

For real-valued processes, the same joint law can be described by its joint cumulative distribution function \(F_{t_1,\ldots,t_k}(x_1,\ldots,x_k)=\mathbb{P}(X_{t_1}\le x_1,\ldots,X_{t_k}\le x_k)\). For discrete states, a table of joint point probabilities is often more direct. In either case, marginal probabilities alone do not generally determine the finite-dimensional distribution: dependence between the selected values matters.

Two Properties That Help Organize the Laws

A finite-dimensional distribution is a pushforward probability law: it is obtained by applying the map \(\omega\mapsto(X_{t_1}(\omega),\ldots,X_{t_k}(\omega))\) to the original probability space. This observation gives both a practical computation method and a consistency check.

Proof. Write \(Y=(X_{t_1},\ldots,X_{t_k})\), and let \(J\) select some of its coordinates in their original order. For any measurable set \(C\) in the space of the selected coordinates, the event that the projection of \(Y\) lies in \(C\) is exactly the event that the selected process values lie in \(C\). Thus its probability is the probability assigned to the inverse image of \(C\) under the projection, which is the projected joint law. In the discrete case, fix values for the selected coordinates. The event that those values occur is the disjoint union of the events obtained by assigning every possible combination of values to the omitted coordinates. Countable additivity says its probability is the sum of the corresponding joint point probabilities. \(\square\)

There is a companion rule for changing coordinate order: if the indices are permuted, the vector’s coordinates are permuted by the same rule, and probabilities are assigned to the correspondingly permuted sets. Both rules follow directly from what the vector records; they do not require independence or stationarity.

Worked Distributions

Why Joint Probabilities Matter

The first example has independent selected coordinates, but that is not automatic for a process. In the threshold example, \(X_0=1\) implies \(X_2=1\), because \(U\le\tfrac14\) also implies \(U\le\tfrac34\). A joint law makes such dependence visible. It also supports calculations of events involving several times, such as the probability that a process visits a specified set of states at two selected indices.

A useful general result underlies the examples: probabilities of joint values are found by identifying the underlying outcomes that produce those values. For a discrete state space, this gives a point-mass formula.

Proof. For each \(x=(x_1,\ldots,x_k)\in S^k\), let \(A_x=\{\omega:(X_{t_1}(\omega),\ldots,X_{t_k}(\omega))=x\}\). If \(x\ne y\), then at least one coordinate differs, so \(A_x\cap A_y=\varnothing\). The event that the vector belongs to \(B\) is exactly the union of \(A_x\) over \(x\in B\). The union is countable because \(S\) is countable and \(k\) is finite. Countable additivity therefore gives \(\mathbb{P}((X_{t_1},\ldots,X_{t_k})\in B)=\sum_{x\in B}\mathbb{P}(A_x)\), which is the stated formula. Taking \(B=S^k\) also shows that all point masses sum to 1. \(\square\)

In practice, the formula can be implemented by a table: list the possible underlying outcomes, record the process vector for the selected indices, and add probabilities whenever different underlying outcomes give the same vector. If the state space is countable, there may be infinitely many rows, but the same disjoint-event argument applies.

Common Mistake: Multiplying Marginals Without Independence

A frequent error is to compute a joint probability by multiplying two marginal probabilities without first establishing independence. For the threshold process above, the marginal probabilities are \(\mathbb{P}(X_0=1)=\tfrac14\) and \(\mathbb{P}(X_2=1)=\tfrac34\). The incorrect calculation would be \(\mathbb{P}(X_0=1,X_2=1)=\tfrac14\cdot\tfrac34=\tfrac{3}{16}\). But the joint event is exactly \(\{U\le\tfrac14\}\), so its probability is \(\tfrac14\), not \(\tfrac{3}{16}\). The product rule for joint probabilities requires independence of the two random variables; here they are dependent because \(X_0=1\) forces \(X_2=1\).

When computing a finite-dimensional distribution, keep the selected values together as a vector and work with the event that the whole vector takes each value. Then check that the probabilities sum to one and that summing over omitted coordinates recovers the appropriate lower-dimensional distributions. Those checks can expose a missing outcome, a duplicated case, or an incorrect independence assumption.