Thinning & Superposition

Splitting a Poisson process by independent labels, or merging independent Poisson processes, stays Poisson.

1. The idea

Thinning: if each event of a Poisson $(λ)$ process is independently labeled "type A" with probability $p$ and "type B" with probability $1 - p$ , then the type-A events alone form a Poisson process with rate $λ p$ , and type-B events form an independent Poisson process with rate $λ (1 - p)$ .

Superposition: merging independent Poisson processes with rates $λ_{1}, \dots, λ_{k}$ produces a single Poisson process with rate $λ_{1} + \dots + λ_{k}$ .

These two operations are inverses of each other and let you decompose complex arrival streams (e.g. buy vs. sell orders) into simpler independent pieces, or combine several sources into one.

2. Why it matters

Order flow is naturally split into buy/sell or by venue; thinning and superposition let you move fluidly between an aggregate arrival rate and its independent components without re-deriving distributions from scratch.

3. Core mathematics

Thinning

N (t) \sim Poisson (λ t) \Rightarrow N_{A} (t) \sim Poisson (λ p t), N_{B} (t) \sim Poisson (λ (1 - p) t)

Independence after thinning

N_{A} (t) ⊥ N_{B} (t)

the split streams are independent of each other

Superposition

N_{1} (t) + \dots + N_{k} (t) \sim Poisson ((i \sum λ_{i}) t)

for independent Poisson processes

4. How to recognize it

Think of this when…

Each event is independently classified into types (success/failure, buy/sell) after a Poisson arrival
"What fraction of arrivals are of type X" combined with a rate
Merging multiple independent arrival streams into one aggregate process
Questions asking for the joint or marginal distribution of split event counts

5. Worked example

Worked example

Orders arrive as Poisson $(λ = 10)$ per minute; each is independently a buy with probability $0.6$ or a sell with probability $0.4$ . Find the probability of exactly $4$ buy orders in one minute, and the combined rate if a second independent order stream with rate $5$ /min is added.

1
By thinning, buy orders form Poisson $(10 \times 0.6) = Poisson (6)$ per minute.
2
Apply the Poisson PMF for exactly 4 buys.
$P (N_{A} (1) = 4) = \frac{e ^{- 6} 6 ^{4}}{4 !} = \frac{1296 e ^{- 6}}{24} \approx 0.1339$
3
By superposition, adding the independent stream of rate 5 gives a combined rate.
$λ_{total} = 10 + 5 = 15 per minute$
Answer: $P (4 buys) \approx 0.134$ ; combined rate after superposition is $15$ /min.

6. Common mistakes

Assuming thinned streams are dependent because they came from the same original process (they are independent)
Forgetting to multiply the base rate by $p$ (or $1 - p$ ) before applying the Poisson formula
Applying superposition to processes that are not independent
Confusing thinning (splitting one process by type) with taking a subset of time (which is just stationary increments)

7. Quick check

A Poisson $(8)$ /hr process is thinned with $p = 0.25$ . What is the rate of the resulting sub-process?

Two independent Poisson processes with rates $3$ and $7$ per minute are merged. What is the merged rate?

8. Practice

2 QuantSolve questions use this concept.

Practice this concept

9. Related methods

Poisson Thinning

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