Why Jump Structure Matters
A counting process summarizes how many events have occurred by each time. Its path contains more information than the final count: it also records when the count increases and by how much. In event-history data, distinguishing a single event at a time from a batch of events is essential to describing the process correctly. A process with only unit jumps has a particularly useful representation in terms of an ordered sequence of event times.
The focus here is the path structure of a counting process. It is not a model for the probability distribution of event times, nor does the definition by itself imply a particular event rate or independence assumption.
The distinction between a jump size and an increment over an interval is important. At a time with no jump, the jump size is zero; at a jump time, its size is a positive integer. For \(0\leq a<b\), the increment \(N(b)-N(a)\) counts events in \((a,b]\). It can be zero even for a simple process, and can exceed one if that interval contains several separate unit jumps.
Event Times and the Simple-Process Representation
For each positive integer \(k\), define the \(k\)th event time as the first time the count reaches at least \(k\). If the count never reaches \(k\), set that event time to infinity.
For a simple process, the finite event times satisfy \(T_1<T_2<T_3<\cdots\): distinct events cannot occur at the same time. At any fixed \(t\), the count is the number of these event times that have occurred by \(t\).
The sum has only finitely many nonzero terms at each finite \(t\), by the assumption of finitely many events on bounded intervals. More generally, if a process permits a jump of size greater than one, its event-occurrence times defined by the threshold rule can repeat. The sum of indicators still counts occurrences, but repeated times encode multiple events at the same instant; the process is then not simple.
Worked Example: Reconstructing a Simple Path From Event Times
Suppose a hypothetical subject has event times \(T_1=2\), \(T_2=5\), and \(T_3=9\) days, with no further events. Find the count at \(t=4\), \(t=5\), and \(t=10\), and check the jump sizes.
Use the event-time representation \(N(t)=\sum_k\mathbf{1}\{T_k\leq t\}\). At \(t=4\), only the first event time is at or before 4, so \(N(4)=1+0+0=1\). At \(t=5\), the first two event times are included, so \(N(5)=1+1+0=2\). At \(t=10\), all three are included, so \(N(10)=1+1+1=3\).
The path is constant between the listed event times and rises by one at days 2, 5, and 9. Thus \(\Delta N(2)=\Delta N(5)=\Delta N(9)=1\), and there are no other positive jumps. The event times are distinct, so this is a simple counting process.
Why the Representation Holds
The event-time formula is a direct consequence of what the \(k\)th event time means. A careful argument also shows why simplicity corresponds to distinct event times.
At a fixed time \(t\), the indicator \(\mathbf{1}\{T_k\leq t\}\) equals one exactly when the count has reached level \(k\) by time \(t\).
If \(N(t)=m\), the reached levels are \(1,\ldots,m\), so exactly \(m\) indicators equal one and all indicators for \(k>m\) equal zero. Their sum is \(m=N(t)\).
If \(N\) has a jump of size \(r>1\) at time \(u\), it crosses \(r\) successive integer thresholds at \(u\), so the corresponding event-occurrence times tie. Conversely, if successive threshold times tie at \(u\), the count crosses at least two thresholds at that time, so \(\Delta N(u)\geq2\). Therefore, all jumps have size one exactly when finite event-occurrence times are strictly increasing.
This argument is pathwise: it applies to each sample path satisfying the counting-process conditions. It requires no distributional assumption about event times. For a simple process, the \(k\)th threshold is crossed one level at a time; for a process that permits multiple events at one instant, several thresholds may be crossed together.
Worked Example: Checking Whether an Observed Path Is Simple
In a hypothetical event log, the cumulative count is 0 before day 1, becomes 1 on day 1, remains 1 through day 3, becomes 2 on day 4, and remains 2 afterward. Determine whether the process is simple.
The count increases from 0 to 1 at day 1, giving \(\Delta N(1)=1-0=1\). It does not increase between days 1 and 4. At day 4, it increases from 1 to 2, giving \(\Delta N(4)=2-1=1\). Every positive jump has size one, so this path is simple. Its event times are \(T_1=1\) and \(T_2=4\), which are distinct. For example, \(N(3)=\mathbf{1}\{1\leq3\}+\mathbf{1}\{4\leq3\}=1+0=1\), and \(N(4)=1+1=2\).
Checking only the final count of 2 would not establish simplicity. The same final count could arise from two unit jumps at separate times or one jump of size two.
When a Process Is Not Simple
A general counting process still has nondecreasing integer-valued paths, but it can allow multiple event occurrences at one time. A jump of size \(r\) means that the count increases by \(r\) at that instant. This may represent a true batch event, multiple events recorded with an identical time, or a data structure that aggregates events. The path alone specifies the jump size, but does not explain why the events coincide.
Worked Example: A Multiple Jump and Its Event-Time Encoding
Suppose a hypothetical process has count 0 before day 3 and count 3 from day 3 onward. Find the jump size and event-occurrence times at day 3, and decide whether the process is simple.
The left limit is \(N(3-)=0\), while \(N(3)=3\). Therefore, \(\Delta N(3)=N(3)-N(3-)=3-0=3\). This is a positive integer jump of size three, not a simple jump.
Under the threshold definition, \(T_1=T_2=T_3=3\), since the count has reached each of the first three thresholds by day 3 and reached none of them before then. Thus \(N(t)=\mathbf{1}\{T_1\leq t\}+\mathbf{1}\{T_2\leq t\}+\mathbf{1}\{T_3\leq t\}\). At \(t=2\), the sum is \(0+0+0=0\); at \(t=3\), it is \(1+1+1=3\), matching the path. The tied event times reflect the multiple jump.
| Feature | Simple counting process | General counting process |
|---|---|---|
| Jump size at a jump time | One | Any positive integer |
| Increment on an interval | Zero or a positive integer, possibly more than one if there are separate jumps | Zero or a positive integer, possibly including multiple events at one jump |
| Event-occurrence times | Finite event times are strictly increasing | Event-occurrence times may repeat |
| Information needed to describe a path | Ordered event times determine the path | Event times with multiplicities, or jump times and jump sizes, determine the path |
The alternatives in the last row are two ways of representing the same count path; neither specifies a probabilistic model. If the scientific question distinguishes types of events, a total count alone may not retain that information. The introductory material on event histories describes the broader role of event records; the path-level distinction here concerns only whether counts rise one at a time.
Diagnostics, Assumptions, and Common Mistakes
To assess simplicity in recorded data, first inspect the cumulative count at every recorded jump time, not merely at a few widely spaced observation times. For each time \(t\) at which the recorded count changes, calculate \(N(t)-N(t-)\). The path is simple if and only if every such difference is one. If the data are stored as event records rather than a cumulative count, check whether two or more event records for the same process share a time.
That check depends on what the recorded timestamps mean. If times have been rounded to the nearest day, two events with the same recorded day might have occurred at different times. Conversely, if a system aggregates multiple events under one timestamp, an observed jump greater than one may reflect recording practice rather than a scientifically meaningful simultaneous occurrence. The observed path can establish the recorded jump size; deciding what that jump represents requires knowledge of the measurement and recording process.
- Confusing an increment with a jump size. An interval increment of 2 does not show a multiple jump: it could result from two distinct unit jumps within the interval. Inspect the path at the event times.
- Allowing a zero-sized jump. A time with no jump has \(\Delta N(t)=0\), but it is not a jump time. At a jump time, the size is positive; it is one for a simple process and may be greater than one for a general process.
- Inferring simplicity from an event-time list with coarse timestamps. Equal recorded times suggest a tie in the recorded data, but do not by themselves prove true simultaneity if timestamps are rounded.
- Assuming the representation gives a probability model. The event-time sum reconstructs a path. It does not specify event-time distributions, dependence, or rates.
A final limitation is that a single counting process records only the total number of events over time. It does not, by itself, distinguish different event types or transitions. When that distinction matters, a richer representation is needed; the count and jump facts established here remain useful building blocks.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A process has a jump from 4 to 5 at time \(t=6\). What is its jump size, and is this jump compatible with simplicity?
- A simple process has two event times at days 2 and 7. Find the count at day 6 and the increment \(N(8)-N(6)\).
- A process jumps from 0 to 2 at time 4. What are its first two event-occurrence times, and is the process simple?
- Over the interval \((1,5]\), the count increases by 2. Does that alone establish a multiple jump? Explain.
- Two event records share a recorded date, but the system rounds times to the nearest day. What can and cannot be concluded about simplicity?
Solutions. (1) The jump size is \(5-4=1\), so this jump is compatible with simplicity. The whole process is simple only if every other jump also has size one. (2) By day 6, only the event at day 2 has occurred, so \(N(6)=1\). By day 8 both events have occurred, so \(N(8)=2\), and \(N(8)-N(6)=2-1=1\). (3) Both thresholds are first reached at day 4, so \(T_1=T_2=4\). The jump size is 2, and the process is not simple. (4) No. The increment of 2 could come from two separate unit jumps in \((1,5]\), or from a single jump of size 2. The path at the jump times is needed to distinguish them. (5) The records have the same rounded date, but the events might have occurred at different times within that day. The recorded timestamps alone do not establish whether the underlying process had a multiple jump; higher-resolution times or information about the recording rule is needed.