Two Different Questions: When and What?
A stochastic process has two kinds of structure. Its index set specifies which labels, times, or locations are used to index its random variables. Its state space specifies the set in which the random variables take their values. These answer different questions: When or where is the process indexed? and What values can it take? Keeping the two answers separate prevents a common classification error.
Formally, on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), a process with index set \(T\) and state space \(S\) is a family \((X_t)_{t\in T}\) of random variables \(X_t:\Omega\to S\). Each \(X_t\) is measurable with respect to the relevant sigma-algebra on \(S\). In this tutorial, a discrete set means a finite or countably infinite set; a continuous set means an uncountable set such as an interval in \(\mathbb{R}\) or \(\mathbb{R}\) itself. Some index sets, such as a union of a countable set and an interval, do not fit neatly into either label; describe them as mixed rather than forcing a classification.
The state space is a declared set containing the process values. It need not be the smallest such set: a process taking only the values \(0\) and \(1\) can be viewed as taking values in \(\mathbb{R}\), though \(\{0,1\}\) is usually the more informative state space. When a smallest natural state space matters, one can specify the set of values attained over the process or a suitable support; these notions should not be confused with the index set.
The Four Basic Combinations
For the common cases, index type and state-space type combine independently:
| Index set \(T\) | State space \(S\) | Typical description |
|---|---|---|
| Discrete | Discrete | A sequence of categorical or count-valued observations |
| Discrete | Continuous | A sequence of real-valued measurements |
| Continuous | Discrete | A time-indexed count or a continuous-location label field |
| Continuous | Continuous | A real-valued process indexed by time or spatial location |
Here, “continuous” does not mean that sample paths are continuous. A process indexed by all \(t\ge0\) has a continuous index set, whether or not its values vary continuously with \(t\). Likewise, a process with real-valued states can jump between values or take values only in a countable subset. Questions about sample-path behavior are separate from this classification.
Proposition (independent classification). A process is classified by two choices: the type of its index set and the type of its state space. Knowing one of these types alone does not determine the other.
Proof. By definition, the process is a family \((X_t)_{t\in T}\) of maps into \(S\). The set \(T\) supplies the indices and the set \(S\) contains the values. Neither definition imposes a condition on the size or type of the other set. For instance, the definition permits \(T=\mathbb{N}_0\) with \(S=\mathbb{R}\), and it permits \(T=[0,1]\) with \(S=\{0,1\}\). Thus the two classifications must be made separately. \(\square\)
Worked Examples of Index and State Choices
Worked Example: A Categorical Sequence
Let \(\Omega=\{0,1\}\), let \(\mathcal{F}\) be its power set, and assign probabilities \(\mathbb{P}(\{1\})=\tfrac13\) and \(\mathbb{P}(\{0\})=\tfrac23\). For each \(n\in\mathbb{N}_0=\{0,1,2,\ldots\}\), set \(X_n(\omega)=\omega\). Classify this process.
The index set is \(\mathbb{N}_0\), which is countably infinite, so the index is discrete. Each \(X_n\) takes values in \(S=\{0,1\}\), a finite state space. The process is therefore discrete-index / discrete-state. Its values can also be regarded as two categories, such as “absent” and “present.” The fact that the values are categories does not change the index set.
Check: For every \(n\), \(\mathbb{P}(X_n=1)=\mathbb{P}(\{1\})=\tfrac13\), and \(\mathbb{P}(X_n=0)=\mathbb{P}(\{0\})=\tfrac23\). These probabilities sum to \(\tfrac13+\tfrac23=1\), and both states belong to the declared state space.
Worked Example: Continuous Time with Two States
Let \(\Omega=[0,1]\) with its Borel sets and uniform probability measure. For \(t\in[0,1]\), define \(X_t(\omega)=\mathbf{1}_{\{\omega\le t\}}\). Find the index and state types and calculate \(\mathbb{P}(X_{1/3}=1)\).
The index set \(T=[0,1]\) is an interval, hence continuous, while \(S=\{0,1\}\) is finite, hence discrete. At \(t=\tfrac13\), the event \(X_t=1\) is \(\{\omega\in[0,1]:\omega\le\tfrac13\}=[0,\tfrac13]\). Its uniform probability is its length, \(\tfrac13\). Thus this is a continuous-index / discrete-state process.
Check: At \(t=0\), the event is \(\{\omega=0\}\), which has probability \(0\); at \(t=1\), every \(\omega\in[0,1]\) satisfies \(\omega\le1\), so the probability is \(1\). In both cases \(X_t\) still takes values in \(\{0,1\}\), and the index set remains the interval \([0,1]\).
Worked Example: A Spatial Field on a Lattice
Use \(\Omega=[0,1)\) with uniform probability measure and index the process by the two-dimensional integer lattice \(T=\mathbb{Z}^2\). For \(k=(m,n)\in T\), define \(X_{(m,n)}(\omega)=m+\omega\). Classify the process and calculate \(\mathbb{P}(X_{(2,-1)}<\tfrac52)\).
The lattice \(\mathbb{Z}^2\) is countable, so its index set is discrete; it represents spatial locations on a grid. For each integer \(m\), the values at indices whose first coordinate is \(m\) fill \([m,m+1)\). Taking the union over all integers \(m\) gives \(\mathbb{R}\), so \(S=\mathbb{R}\) is a natural continuous state space. The process is discrete-index / continuous-state.
At index \((2,-1)\), the definition gives \(X_{(2,-1)}=2+\omega\). The strict inequality \(2+\omega<\tfrac52\) is equivalent to \(\omega<\tfrac12\). The event is \([0,\tfrac12)\), whose uniform probability on \([0,1)\) is \(\tfrac12\).
Check: The interval \([0,\tfrac12)\) has length \(\tfrac12\), so the calculated probability is \(\tfrac12\). The strict endpoint matters for describing the event, although including or excluding the single point \(\tfrac12\) does not change its uniform probability.
Worked Example: A Continuous Spatial Index and Continuous Values
Let \(\Omega=[0,1]\) with uniform probability measure, let \(T=[0,1]\) represent spatial locations, and set \(X_x(\omega)=x+\omega\). Classify the process and calculate \(\mathbb{P}(X_{1/4}\le\tfrac34)\).
The index set is the continuous interval \([0,1]\). For a fixed \(x\), the values lie in \([x,x+1]\); across all \(x\in[0,1]\), the values fill \([0,2]\). Indeed, \(x+\omega\ge0\) and \(x+\omega\le2\), and every \(y\in[0,2]\) can be represented as \(y=x+\omega\) with \(x,\omega\in[0,1]\). Thus \(S=[0,2]\) is a continuous state space, and the pair is continuous-index / continuous-state.
At \(x=\tfrac14\), the event \(X_x\le\tfrac34\) is \(\tfrac14+\omega\le\tfrac34\), or \(\omega\le\tfrac12\). Under the uniform distribution on \([0,1]\), this event has probability \(\tfrac12\).
Check: The event is the interval \([0,\tfrac12]\), of length \(\tfrac12\). The process values are within \([0,2]\), since the minimum possible sum is \(0+0=0\) and the maximum is \(1+1=2\).
Changing Indices or Values
Descriptions can change when a process is reindexed or its values are recoded. These operations affect different parts of the classification.
Proof. For reindexing, each \(u\in T'\) is sent to an index \(\phi(u)\in T\), so \(Y_u=X_{\phi(u)}\) still takes values in \(S\). The labels used for the new family are exactly \(u\in T'\), so its index set is \(T'\). For recoding, for every \(t\in T\), \(X_t(\omega)\in S\); applying \(f\) gives \(f(X_t(\omega))\in f(S)\subseteq S'\). The labels \(t\) have not changed, so the index set remains \(T\). These conclusions follow directly from the definitions of the two operations. \(\square\)
If \(f\) is many-to-one, recoding can collapse distinct states into one; the new process may have a smaller natural state space. If \(f\) is one-to-one, distinct values remain distinct, although the names or numerical representation of those values may change. A bijection used to reindex changes the labels but not the values attached through that relabeling.
Choosing the Right Description
“Sequence” and “time-indexed process” are usually descriptions of the index set: a sequence is indexed by discrete steps, while a process indexed by a time interval uses a continuous index set. “Spatial process” says that indices represent locations; those locations may form a discrete grid or a continuous region. “Categorical” describes the states, not the indices. A categorical sequence has discrete indices and categorical values, but a categorical process can also be indexed by continuous time or by locations.
One useful routine is to write the domain and codomain before applying a label: \(t\in T\), \(X_t\in S\). Then ask separately whether \(T\) is finite, countably infinite, an interval, or mixed, and whether \(S\) is finite/countable or uncountable. This also guards against inferring more than the classification says. A continuous index does not establish continuous sample paths, and a continuous state space does not establish a density or any particular distribution.
Common mistake. Suppose \(X_n\) is a sequence taking real values. A wrong classification is “continuous-index / continuous-state” because the values are real numbers. This confuses the two sets. The index is \(n\in\mathbb{N}_0\), which is countable, while the state space can be \(\mathbb{R}\), which is continuous. The correct pair is discrete-index / continuous-state. Conversely, a process indexed by \(t\in[0,1]\) and taking only values in \(\{0,1\}\) has continuous indices and discrete states, as the time-indexed example demonstrates.
Check Your Understanding
For each question, identify the index set and state space separately before giving the classification.
- A sequence \((X_n)_{n\ge0}\) takes values in \(\{A,B,C\}\). What is its index–state pair, and which part of the description makes it categorical?
- A process is indexed by all \(t\ge0\) and takes values in \(\mathbb{N}_0\). Which set is continuous, and which is discrete?
- A real-valued field is indexed by \(\mathbb{Z}^2\). Is its index set discrete or continuous? Does the real-valued state space change that answer?
- If \(Y_t=f(X_t)\) for a many-to-one map \(f\), which part of the process classification can change?
- A process is indexed by a spatial region \([0,1]\) and takes values in \(\{0,1\}\). Give its pair and explain why the word “spatial” alone does not determine the state type.