Independent Increments

What Independent Increments Means

A process records values over time, but many useful properties concern the changes between times. Independent increments means that changes measured over separate, non-overlapping time intervals are independent random variables. The condition applies to every finite collection of such intervals; checking only two intervals is not enough to establish the property in general.

Let \((X_t)_{t\ge 0}\) be a real-valued process on a probability space \((\Omega,\mathcal{F},\mathbb{P})\), with time index \(t\ge 0\) and state space \(S\subseteq\mathbb{R}\). For \(0\le s<t\), the increment over \((s,t]\) is \(X_t-X_s\). The convention \((s,t]\) lets adjacent intervals, such as \((s,t]\) and \((t,u]\), be treated as disjoint. In discrete time, the same definition applies with integer endpoints \(s,t\).

The word mutually matters: the definition requires the joint factorization for any finite number of increments, not merely pairwise independence. The condition concerns increments, not the values \(X_{t_1},\ldots,X_{t_m}\). Even when a process has independent increments, its values at different times can be dependent because they may share earlier changes.

Independent increments do not require the increments to have the same distribution. For example, separate intervals may contain sums of independent variables with different distributions, while those sums remain independent. Whether increment distributions depend on interval location or length is a separate question; independence alone makes no such claim.

Why Partial Sums Have Independent Increments

Start with a sequence of independent random variables \((Y_k)_{k\ge1}\) on \((\Omega,\mathcal{F},\mathbb{P})\), and define the discrete-time partial-sum process by \(S_0=0\) and \(S_n=\sum_{k=1}^n Y_k\) for \(n\ge1\). Its state space can be taken to be \(\mathbb{R}\). An increment from time \(a\) to time \(b\), where \(0\le a<b\) are integers, is exactly the sum of the variables with indices \(a+1,\ldots,b\).

Proof. Choose any finite collection of integer intervals with \(0\le a_1<b_1\le a_2<b_2\le\cdots\le a_m<b_m\). For each \(r\), the increment \(S_{b_r}-S_{a_r}\) equals \(\sum_{k=a_r+1}^{b_r}Y_k\). The index sets \(\{a_r+1,\ldots,b_r\}\) are pairwise disjoint: the next one begins at \(a_{r+1}+1\), which is greater than \(b_r\) when \(a_{r+1}\ge b_r\). Thus each increment is a function only of the variables in its own index set. To verify independence, take Borel sets \(A_1,\ldots,A_m\). The event \(\{S_{b_r}-S_{a_r}\in A_r\}\) is determined by the variables in the \(r\)-th index set. Independence of the sequence gives independence of the subvectors formed from these disjoint sets of variables, and therefore the probability of the intersection of these events is the product of their probabilities. This is the definition of mutual independence of the increments. \(\square\)

A Counting Process with Independent Increments

A counting process records the number of events observed by a given time. For a simple model, let \((Y_k)_{k\ge1}\) be independent Bernoulli variables on \((\Omega,\mathcal{F},\mathbb{P})\), each with success probability \(p\), where \(0<p<1\). Interpret \(Y_k=1\) as an event at integer time \(k\), and \(Y_k=0\) as no event then. Define the continuous-time process \(N_t=\sum_{k=1}^{\lfloor t\rfloor}Y_k\) for \(t\ge0\), with state space \(S=\{0,1,2,\ldots\}\); an empty sum is \(0\). It starts at \(N_0=0\), is nondecreasing, and counts events at integer times no later than \(t\).

Proof. Fix any finite collection of disjoint time intervals \((s_r,t_r]\) in chronological order, with \(0\le s_1<t_1\le s_2<t_2\le\cdots\le s_m<t_m\). By the definition of \(N_t\),

If \(\lfloor s_r\rfloor=\lfloor t_r\rfloor\), this sum is empty and the increment is the constant \(0\), which is independent of every random variable. Otherwise, the sum uses the indicators whose integer times lie in \((s_r,t_r]\). Since \(t_r\le s_{r+1}\), the index sets used for consecutive intervals do not overlap: every index in the first is at most \(\lfloor t_r\rfloor\), while an index in the next is at least \(\lfloor s_{r+1}\rfloor+1\ge\lfloor t_r\rfloor+1\). The same separation holds for any pair of intervals in the collection. Each increment is a function of its own group of independent indicators, so these increments are mutually independent. This verifies the definition for every finite collection of disjoint intervals. \(\square\)

Disjoint Intervals Are Not Enough by Themselves

Disjoint time intervals identify which increments must be compared; they do not guarantee that those increments are independent. The process must have the appropriate probability structure. In the partial-sum and Bernoulli-count examples, independence followed because each interval used a separate group of independent variables. A process that reuses the same random input can fail the condition.

Common mistake. A tempting but incorrect argument is: “The intervals \((0,1]\) and \((1,2]\) are disjoint, so their increments are independent.” Disjointness is a hypothesis for applying the independent-increments property, not a proof that a given process has it. In the example, the incorrect calculation would replace \(\mathbb{P}(A=1,B=1)=1/2\) by the product \(1/4\). The correction is to calculate the joint probability from the process itself and compare it with the product of the marginals. The mismatch shows dependence.

To test a proposed process, write each increment in terms of the underlying random variables. If different intervals produce functions of disjoint groups from an independent family, the independence argument is available. If the groups overlap or share a random input, independence does not follow; calculate the joint and marginal probabilities or distributions to decide the question. This method also keeps separate two claims that are easy to confuse: independent increments is a property of changes across disjoint intervals, while independence of process values is a different property.