A counting process possesses independent increments if the number of events between s and t is independent of the number between t and t+u for all u>0.

A counting process possesses independent increments if the number of events between s and t is independent of the number between t and t+u for all u>0.

• A. True
• B. False
• C. Sometimes
• D. Never

Answer: (A)

Independent increments means the counts in non-overlapping time intervals are independent. For a counting process N(t), the number of events between s and t is N(t) − N(s) and between t and t+u is N(t+u) − N(t). These correspond to disjoint intervals [s, t] and [t, t+u], sharing only the boundary. If the process has independent increments, these two quantities are independent for every s < t and every u > 0. That is exactly what the statement asserts, so the statement is true. If a process lacked independent increments, this independence could fail, but the property as stated is the defining behavior for counting processes with independent increments.

Independent increments means the counts in non-overlapping time intervals are independent. For a counting process N(t), the number of events between s and t is N(t) − N(s) and between t and t+u is N(t+u) − N(t). These correspond to disjoint intervals [s, t] and [t, t+u], sharing only the boundary. If the process has independent increments, these two quantities are independent for every s < t and every u > 0. That is exactly what the statement asserts, so the statement is true. If a process lacked independent increments, this independence could fail, but the property as stated is the defining behavior for counting processes with independent increments.