On Continuous Versions of Poisson and Binomial Distributions

On Continuous Versions of Poisson and Binomial Distributions Andrii Ilienko National Technical University of Ukraine, Kiev January 12, 2011

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Problem statement Problem statement

Let πλ be the Poisson measure:

Integral representations

supp πλ = {0, 1, 2, . . . }, Continuous versions of πλ and βN,p


e−λ λk , πλ {k} = k!

Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

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Problem statement Problem statement

Let πλ be the Poisson measure:

Integral representations

supp πλ = {0, 1, 2, . . . }, Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process


e−λ λk , πλ {k} = k!

βN,p be the binomial measure: supp πλ = {0, 1, 2, . . . , N},

βN,p

!


N k p (1 − p)N−k . {k} = k

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Problem statement Problem statement

Let πλ be the Poisson measure:

Integral representations

supp πλ = {0, 1, 2, . . . }, Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process


e−λ λk , πλ {k} = k!

βN,p be the binomial measure: supp πλ = {0, 1, 2, . . . , N},

βN,p

!


N k p (1 − p)N−k . {k} = k

The central aim of the talk is consideration of absolutely continuous versions of πλ and βN,p .

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Problem statement Integral representations

In other words, we spread measures πλ and βN,p continuously onto [0, ∞) and [0, N + 1], respectively.

Continuous versions of πλ and βN,p Limit theorem

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2

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Moments of continuous Poisson distribution An application to the Γ-process

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Problem statement Integral representations

In other words, we spread measures πλ and βN,p continuously onto [0, ∞) and [0, N + 1], respectively.

Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution

0

1

2

3

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How to spread NATURALLY? And what is NATURALLY?

An application to the Γ-process

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Problem statement Integral representations

In other words, we spread measures πλ and βN,p continuously onto [0, ∞) and [0, N + 1], respectively.

Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

0

1

2

3

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5

How to spread NATURALLY? And what is NATURALLY? We use integral representations of Fπ (·) and Fβ (·) via complete and incomplete Euler Γ- and B-functions.

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Integral representations Problem statement Integral representations Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

Recall classical definitions: R∞

−t x−1 e t dt, x > 0; 0 R∞ incomplete Γ-function: Γλ (x) = λ e−t t x−1 dt, x > 0; R1 B-function: B(x, y) = 0 t x−1 (1 − t)y−1 dt, x, y > 0; R1 incomplete B-function: B p (x, y) = p t x−1 (1 − t)y−1 dt, x, y > 0;

Γ-function: Γ(x) =

here λ > 0, 0 < p < 1.

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Problem statement Integral representations Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

Lemma 1 (Probabilistic folklore). Distribution functions    

Fπ ≔ πλ (−∞, x) , x ∈ R and Fβ ≔ βN,p (−∞, x) , x ∈ R admit following representations:


Γλ ⌈x⌉ Fπ (x) =
· 1{x>0} , Γ ⌈x⌉
B p ⌈x⌉, N + 1 − ⌈x⌉ Fβ (x) =
· 1{0N} . B ⌈x⌉, N + 1 − ⌈x⌉

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Problem statement Integral representations Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

Lemma 1 (Probabilistic folklore). Distribution functions    

Fπ ≔ πλ (−∞, x) , x ∈ R and Fβ ≔ βN,p (−∞, x) , x ∈ R admit following representations:


Γλ ⌈x⌉ Fπ (x) =
· 1{x>0} , Γ ⌈x⌉
B p ⌈x⌉, N + 1 − ⌈x⌉ Fβ (x) =
· 1{0N} . B ⌈x⌉, N + 1 − ⌈x⌉

Proof is a simple calculus.

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Continuous versions of πλ and βN,p Problem statement Integral representations

Lemma 1 makes it possible to introduce the NATURAL continuous counterparts of discrete measures πλ and βN,p .

Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

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Continuous versions of πλ and βN,p Problem statement Integral representations Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution

Lemma 1 makes it possible to introduce the NATURAL continuous counterparts of discrete measures πλ and βN,p . Definition 1. By continuous Poisson distribution with parameter λ > 0 we will mean the probabilistic measure π˜ λ with
Γλ (x) Fπ˜ (x) ≔ π˜ λ (−∞, x) = · 1{x>0} , Γ(x)

x ∈ R.

An application to the Γ-process

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Continuous versions of πλ and βN,p Problem statement Integral representations Continuous versions of πλ and βN,p Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

Lemma 1 makes it possible to introduce the NATURAL continuous counterparts of discrete measures πλ and βN,p . Definition 1. By continuous Poisson distribution with parameter λ > 0 we will mean the probabilistic measure π˜ λ with
Γλ (x) Fπ˜ (x) ≔ π˜ λ (−∞, x) = · 1{x>0} , Γ(x)

x ∈ R.

Definition 2. By continuous binomial distribution with parameters y > 0, 0 < p < 1 we will mean the probabilistic measure β˜ N,p with
B p (x, y + 1 − x) ˜ · 1{0 0. 0

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Problem statement Integral representations Continuous versions of πλ and βN,p

Theorem 4. The Laplace transform of mk has the form: k! m ˆ k (s) = , k s ln (1 + s)

s > 0.

Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

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Problem statement Integral representations

Theorem 4. The Laplace transform of mk has the form: k! m ˆ k (s) = , k s ln (1 + s)

Continuous versions of πλ and βN,p

s > 0.

Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

The double Laplace transform of the family (˜πλ , λ > 0) is (Laplace-Stieltjes w.r.t. measure and Laplace w.r.t. λ) ϕ(p, ˆ s) ≔

"

−px−sλ

e (0,∞)×(0,∞)

1 ln(1 + s) π˜ λ (dx) dλ = , s p + ln(1 + s) p, s > 0.

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An application to the Γ-process Problem statement Integral representations Continuous versions of πλ and βN,p


Consider Γ-process X = X(t), t ≥ 0 , i.e. L´evy process with the transition density βαt αt−1 −βx ft (x) = x e , Γ(αt)

x ≥ 0.

Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

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An application to the Γ-process Problem statement Integral representations Continuous versions of πλ and βN,p


Consider Γ-process X = X(t), t ≥ 0 , i.e. L´evy process with the transition density βαt αt−1 −βx ft (x) = x e , Γ(αt)

x ≥ 0.

Limit theorem Moments of continuous Poisson distribution An application to the Γ-process

The continuous Poisson distribution π˜ λ finds a use for study of the process X. Theorem 5. Let τc be the hitting time of the level c > 0. Then the r.v. ατc has the π˜ βc -distribution.

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