On a limit theorem for a non-linear scaling

Zbigniew J. Jurek

arXiv:1906.10522·math.PR·May 23, 2022

On a limit theorem for a non-linear scaling

Zbigniew J. Jurek

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TL;DR

This paper establishes a limit theorem for sums of i.i.d. positive random variables scaled by a non-linear transform, showing the limits are either degenerate or compound Poisson distributions.

Contribution

It introduces a novel non-linear scaling method and characterizes the possible weak limit distributions for sums of i.i.d. variables under this transformation.

Findings

01

Weak limits are either degenerate or compound Poisson distributions.

02

The non-linear transform used is $ ext{max}(0, x - r)$.

03

The result extends classical limit theorems to non-linear scaling contexts.

Abstract

In this note, we proved that weak limits, of sums of independent positive identically distributed random variables which are re-normalized by a non-linear shrinking transform $max (0, x - r)$ , are either degenerate or (some) compound Poisson distributions.

Equations64

U_{r} : [0, \infty) \to [0, \infty) \mbox b y U_{r} (x) := max {0, x - r} \equiv (x - r)_{+}

U_{r} : [0, \infty) \to [0, \infty) \mbox b y U_{r} (x) := max {0, x - r} \equiv (x - r)_{+}

U_{r} (U_{s} (x)) = U_{r + s} (x) \mbox an d ∣ U_{r} (x) - U_{s} (x) ∣ \leq ∣ r - s ∣, \mbox f or 0 \leq x, r, s < \infty.

U_{r} (U_{s} (x)) = U_{r + s} (x) \mbox an d ∣ U_{r} (x) - U_{s} (x) ∣ \leq ∣ r - s ∣, \mbox f or 0 \leq x, r, s < \infty.

S_{n} := U_{r_{n}} (X_{1}) + U_{r_{n}} (X_{2}) + ... + U_{r_{n}} (X_{n}) \Rightarrow S, \mbox a s n \to \infty,

S_{n} := U_{r_{n}} (X_{1}) + U_{r_{n}} (X_{2}) + ... + U_{r_{n}} (X_{n}) \Rightarrow S, \mbox a s n \to \infty,

(i) there exists a constant c > 0 such that S = c with P.1,

(i) there exists a constant c > 0 such that S = c with P.1,

(ii) there exists an a > 0 such that S = d e^{- a} k = 0 \sum \infty \frac{a ^{k}}{k !} ν^{* k},

(ii) there exists an a > 0 such that S = d e^{- a} k = 0 \sum \infty \frac{a ^{k}}{k !} ν^{* k},

P (U_{r} (X) \leq x) = P (X \leq x + r) \mbox f or x \geq 0.

P (U_{r} (X) \leq x) = P (X \leq x + r) \mbox f or x \geq 0.

L (U_{r} (X); t) = 1 - \int_{[0, \infty)} [1 - e^{- t x}] d F (x + r); F (x) := P (X \leq x), x \geq 0

L (U_{r} (X); t) = 1 - \int_{[0, \infty)} [1 - e^{- t x}] d F (x + r); F (x) := P (X \leq x), x \geq 0

L (U_{r} (X); t)] = F (r) + \int_{(0, \infty)} e^{- t x} d F (x + r) = 1 - \int_{(r, \infty)} [1 - e^{- t (y - r)}] d F (y) = 1 - \int_{(0, \infty)} [1 - e^{- t x}] d F (x + r) = 1 - \int_{[0, \infty)} [1 - e^{- t x}] d F (x + r) .

L (U_{r} (X); t)] = F (r) + \int_{(0, \infty)} e^{- t x} d F (x + r) = 1 - \int_{(r, \infty)} [1 - e^{- t (y - r)}] d F (y) = 1 - \int_{(0, \infty)} [1 - e^{- t x}] d F (x + r) = 1 - \int_{[0, \infty)} [1 - e^{- t x}] d F (x + r) .

\mathfrak{L}(S_{n};t)=(\mathbb{E}[e^{-t\,U_{r_{n}}(X_{1})}])^{n}=\Big{(}1-\int_{[0,\infty)}[1-e^{-tx}]dF(x+r_{n})\Big{)}^{n}\\ =\Big{(}1-\frac{1}{n}\,[\,\int_{[0,\infty)}\frac{1-e^{-tx}}{1-e^{-x}}dG_{n}(x)\,]\Big{)}^{n},\ \mbox{where distributions $G_{n}$ are defined as}\ \\ G_{n}(y):=n\int_{0}^{y}(1-e^{-x})dF(x+r_{n}),\ y\geq 0;\ \ \,\,G_{n}(\infty)\leq n(1-F(r_{n})).

\mathfrak{L}(S_{n};t)=(\mathbb{E}[e^{-t\,U_{r_{n}}(X_{1})}])^{n}=\Big{(}1-\int_{[0,\infty)}[1-e^{-tx}]dF(x+r_{n})\Big{)}^{n}\\ =\Big{(}1-\frac{1}{n}\,[\,\int_{[0,\infty)}\frac{1-e^{-tx}}{1-e^{-x}}dG_{n}(x)\,]\Big{)}^{n},\ \mbox{where distributions $G_{n}$ are defined as}\ \\ G_{n}(y):=n\int_{0}^{y}(1-e^{-x})dF(x+r_{n}),\ y\geq 0;\ \ \,\,G_{n}(\infty)\leq n(1-F(r_{n})).

n \to \infty lim G_{n} (y) = n \to \infty lim n \int_{0}^{y} (1 - e^{- x}) λ e^{- λ (x + r_{n})} d x = a \int_{0}^{y} (1 - e^{- x}) λ e^{- λ x} d x = G (y), \mbox f or 0 \leq y < \infty.

n \to \infty lim G_{n} (y) = n \to \infty lim n \int_{0}^{y} (1 - e^{- x}) λ e^{- λ (x + r_{n})} d x = a \int_{0}^{y} (1 - e^{- x}) λ e^{- λ x} d x = G (y), \mbox f or 0 \leq y < \infty.

L [S; t] = exp \int_{0}^{\infty} \frac{1 - e ^{- t x}}{1 - e ^{- x}} G (d x) = exp (- a \int_{0}^{\infty} (1 - e^{- t x}) λ e^{- λ x} d x = exp a [\frac{λ}{λ + t} - 1] .

L [S; t] = exp \int_{0}^{\infty} \frac{1 - e ^{- t x}}{1 - e ^{- x}} G (d x) = exp (- a \int_{0}^{\infty} (1 - e^{- t x}) λ e^{- λ x} d x = exp a [\frac{λ}{λ + t} - 1] .

G(y):=\lim_{n\to\infty}G_{n}(y)=\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}n\int_{0}^{y}(1-e^{-x})\exp(-(x+r_{n})^{2}/2)dx\\ =\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}n\exp(-r_{n}^{2}/2)\int_{0}^{y}(1-\exp(-x))\exp(-x^{2}/2)\exp(-r_{n}x)dx\\ =\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}n\exp(-r_{n}^{2}/2)\int_{0}^{y}\frac{1-\exp(-x)}{x}x\exp(-x^{2}/2)\exp(-r_{n}x)dx\\ \mbox{(using substitution $xr_{n}=:u$ we get) }\\ =\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}\,n\,(r_{n}^{2}\exp(r_{n}^{2}/2))^{-1}\,\int_{0}^{yr_{n}}\frac{1-\exp(-u/r_{n})}{u/r_{n}}\exp(-\frac{u^{2}}{2r_{n}^{2}})u\exp(-u)du\\ =c\lim_{n\to\infty}\int_{0}^{\infty}[\,1_{(0,yr_{n}]}(u)\,\frac{1-\exp(-u/r_{n})}{u/r_{n}}\exp(-\frac{u^{2}}{2r_{n}^{2}}),]\,u\exp(-u)du\\ =c\int_{0}^{\infty}u\exp(-u)du=c,\ \ \mbox{for}\ y>0.

G(y):=\lim_{n\to\infty}G_{n}(y)=\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}n\int_{0}^{y}(1-e^{-x})\exp(-(x+r_{n})^{2}/2)dx\\ =\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}n\exp(-r_{n}^{2}/2)\int_{0}^{y}(1-\exp(-x))\exp(-x^{2}/2)\exp(-r_{n}x)dx\\ =\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}n\exp(-r_{n}^{2}/2)\int_{0}^{y}\frac{1-\exp(-x)}{x}x\exp(-x^{2}/2)\exp(-r_{n}x)dx\\ \mbox{(using substitution $xr_{n}=:u$ we get) }\\ =\lim_{n\to\infty}(\frac{2}{\pi})^{1/2}\,n\,(r_{n}^{2}\exp(r_{n}^{2}/2))^{-1}\,\int_{0}^{yr_{n}}\frac{1-\exp(-u/r_{n})}{u/r_{n}}\exp(-\frac{u^{2}}{2r_{n}^{2}})u\exp(-u)du\\ =c\lim_{n\to\infty}\int_{0}^{\infty}[\,1_{(0,yr_{n}]}(u)\,\frac{1-\exp(-u/r_{n})}{u/r_{n}}\exp(-\frac{u^{2}}{2r_{n}^{2}}),]\,u\exp(-u)du\\ =c\int_{0}^{\infty}u\exp(-u)du=c,\ \ \mbox{for}\ y>0.

L [S; t] = exp \int_{0}^{\infty} \frac{1 - e ^{- t x}}{1 - e ^{- x}} G (d x) = exp (- c t), t \geq 0,

L [S; t] = exp \int_{0}^{\infty} \frac{1 - e ^{- t x}}{1 - e ^{- x}} G (d x) = exp (- c t), t \geq 0,

\mathfrak{L}(S;t)=\exp\big{(}-\lim_{n\to\infty}\int_{[0,\infty)}\frac{1-e^{-tx}}{1-e^{-x}}dG_{n}(x)\big{)},\ \ t\geq 0,

\mathfrak{L}(S;t)=\exp\big{(}-\lim_{n\to\infty}\int_{[0,\infty)}\frac{1-e^{-tx}}{1-e^{-x}}dG_{n}(x)\big{)},\ \ t\geq 0,

\lim_{k\to\infty}\mathfrak{L}(S_{n_{k}};1)=\lim_{k\to\infty}\big{(}1-\frac{G_{n_{k}}(\infty)}{n}\big{)}^{n}=0

\lim_{k\to\infty}\mathfrak{L}(S_{n_{k}};1)=\lim_{k\to\infty}\big{(}1-\frac{G_{n_{k}}(\infty)}{n}\big{)}^{n}=0

L (S; t) = n \to \infty lim L (S_{n}; t) = exp (- \int_{[0, \infty]} \frac{1 - e ^{- t x}}{1 - e ^{- x}} d G (x)) = exp (- \int_{[0, \infty)} \frac{1 - e ^{- t x}}{1 - e ^{- x}} d G (x)) \cdot exp (- G ({\infty})) .

L (S; t) = n \to \infty lim L (S_{n}; t) = exp (- \int_{[0, \infty]} \frac{1 - e ^{- t x}}{1 - e ^{- x}} d G (x)) = exp (- \int_{[0, \infty)} \frac{1 - e ^{- t x}}{1 - e ^{- x}} d G (x)) \cdot exp (- G ({\infty})) .

H_{n} (u) := \int_{1}^{u} \frac{1}{1 - e ^{- x}} d G_{n} (x); H_{n} (u) \to H (u) := \int_{1}^{u} \frac{1}{1 - e ^{- x}} d G (x); 0 < u < \infty, \mbox an d h e n ce G (u) - G (1) = \int_{1}^{u} (1 - e^{- x}) d H (x) .

H_{n} (u) := \int_{1}^{u} \frac{1}{1 - e ^{- x}} d G_{n} (x); H_{n} (u) \to H (u) := \int_{1}^{u} \frac{1}{1 - e ^{- x}} d G (x); 0 < u < \infty, \mbox an d h e n ce G (u) - G (1) = \int_{1}^{u} (1 - e^{- x}) d H (x) .

H (u) = n \to \infty lim H_{n} (u) = n \to \infty lim n [F (u + r_{n}) - F (1 + r_{n})]

H (u) = n \to \infty lim H_{n} (u) = n \to \infty lim n [F (u + r_{n}) - F (1 + r_{n})]

H (u + w) = H (u) + H (1 + w), 0 < u < \infty

H (u + w) = H (u) + H (1 + w), 0 < u < \infty

H(u)=\lim_{n\to\infty}(n+1)[F(u+r_{n+1})-F(1+r_{n+1}]\\ =\lim_{n\to\infty}\Big{[}n\,[F(u+w_{n}+r_{n})-F(1+w_{n}+r_{n})]+[F(u+r_{n+1})-F(1+r_{n+1}]\Big{]}\\ =\lim_{n\to\infty}\big{(}n\,[F(u+w_{n}+r_{n})-F(1+r_{n})]-n[F(1+w_{n}+r_{n})-F(1+r_{n})]\big{)}\\ =H(u+w)-H(1+w),\ \ \mbox{for}\ 0<u<\infty\ \mbox{and}\ \ 0\leq w\leq\infty.\qquad

H(u)=\lim_{n\to\infty}(n+1)[F(u+r_{n+1})-F(1+r_{n+1}]\\ =\lim_{n\to\infty}\Big{[}n\,[F(u+w_{n}+r_{n})-F(1+w_{n}+r_{n})]+[F(u+r_{n+1})-F(1+r_{n+1}]\Big{]}\\ =\lim_{n\to\infty}\big{(}n\,[F(u+w_{n}+r_{n})-F(1+r_{n})]-n[F(1+w_{n}+r_{n})-F(1+r_{n})]\big{)}\\ =H(u+w)-H(1+w),\ \ \mbox{for}\ 0<u<\infty\ \mbox{and}\ \ 0\leq w\leq\infty.\qquad

H (u + k w) = H (u) + k H (1 + w), k \geq 1

H (u + k w) = H (u) + k H (1 + w), k \geq 1

H (u + a) = b^{- 1} H (u) + H (1 + a), 0 < u < \infty.

H (u + a) = b^{- 1} H (u) + H (1 + a), 0 < u < \infty.

H(u)=\lim_{n\to\infty}k_{n}[F(u+r_{k_{n}})-F(1+r_{k_{n}})]\\ =\lim_{n\to\infty}\frac{k_{n}}{n}\big{[}n[F(u+v_{n}+r_{n})-F(1+r_{n})]-n[F(1+v_{n}+r_{n})-F(1+r_{n})]\big{]}.

H(u)=\lim_{n\to\infty}k_{n}[F(u+r_{k_{n}})-F(1+r_{k_{n}})]\\ =\lim_{n\to\infty}\frac{k_{n}}{n}\big{[}n[F(u+v_{n}+r_{n})-F(1+r_{n})]-n[F(1+v_{n}+r_{n})-F(1+r_{n})]\big{]}.

H (u) = co n s t \mbox or H (u) = α (e^{- γ u} - e^{- γ}), α < 0, γ > 0.

H (u) = co n s t \mbox or H (u) = α (e^{- γ u} - e^{- γ}), α < 0, γ > 0.

H (u) = α e^{γ u} + β \mbox w i t h r es t r i c t i o n s H (1) = 0; H (u) < 0, \mbox f or 0 < u < 1;

H (u) = α e^{γ u} + β \mbox w i t h r es t r i c t i o n s H (1) = 0; H (u) < 0, \mbox f or 0 < u < 1;

G (y) = G (1) + \int_{1}^{y} (1 - e^{- x}) (- α γ) e^{- γ x} d x; G (0) = 0.

G (y) = G (1) + \int_{1}^{y} (1 - e^{- x}) (- α γ) e^{- γ x} d x; G (0) = 0.

G (y) = - α γ \int_{0}^{y} (1 - e^{- x}) e^{- γ x} d x; y \geq 0; α < 0, γ > 0.

G (y) = - α γ \int_{0}^{y} (1 - e^{- x}) e^{- γ x} d x; y \geq 0; α < 0, γ > 0.

L (S; t) = exp (- c t) \mbox or L (S; t) = exp (- α) [\frac{γ}{t + γ} - 1], α < 0, γ > 0,

L (S; t) = exp (- c t) \mbox or L (S; t) = exp (- α) [\frac{γ}{t + γ} - 1], α < 0, γ > 0,

(i) f (x + y) = ϕ (y) f (x) + ψ (y) \mbox or (ii) f (x + y) = f (x) + ψ (y)

(i) f (x + y) = ϕ (y) f (x) + ψ (y) \mbox or (ii) f (x + y) = f (x) + ψ (y)

f (x) = α e^{γ x} + c, ϕ (x) = e^{γ x}, ψ (x) = c [1 - e^{γ x}]; (α, γ, c \mbox a r eco n s t an t s) .

f (x) = α e^{γ x} + c, ϕ (x) = e^{γ x}, ψ (x) = c [1 - e^{γ x}]; (α, γ, c \mbox a r eco n s t an t s) .

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Full text

On a limit theorem for a non-linear scaling

Zbigniew J. Jurek (University of Wrocław)

(March 6, 2021)

Abstract. In this note we proved that weak limits, of sums of positive, independent and identically distributed random variables which are re-normalized by a non-linear shrinking transform $\max(0,x-r)$ , are either degenerate or (some) compound Poisson distributions.

Mathematics Subject Classifications(2020): Primary 60E05, 60E07, 60E10.

Key words and phrases: Laplace transform; weak convergence; functional equation.

*Abbreviated title: On a limit theorem for a non-linear scaling *

Addresses:

Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4

50-384 Wrocław, Poland; www.math.uni.wroc.pl/$\sim$zjjurek ;

e-mail: [email protected]

In a probability theory often one describes behaviors of normalized partial sums of random variables or vectors. Most often a normalization of partial sums is done be affine (linear) transforms thus a normalization is applied directly to each summand; cf. Jurek and Mason(1993) or Meerscheart and Scheffler (2001). In this note positive random variables are normalized by a non-linear shrinking transformation (s-operation) and then partial sums are computed. A limit theorem for those sums is proved for non-negative independent and identically distributed variables; Theorem 1. The Laplace transform and some functional equations are the main tools in proofs. [See also a historical note on the s-operations in Section 6.]

1. A non-linear scaling $U_{r}$ and a theorem.

For $r>0$ , let us define a non-linear scalings (shrinking mappings, in short: s-operations)

[TABLE]

which have, among others, the one-parameter semigroup property:

[TABLE]

If $X$ is a non-negative random variable on a fixed probability space $(\Omega,\mathcal{F},P)$ then $U_{r}(X)$ may model a situation when one receives only an excess above some positive level $r$ . Similarly, such formula appears in mathematical finance as European call options where $X$ represents a price of a stock, $r$ is a strike price, and $U_{r}(X)$ is an expected investor’s gain. Hence, sums $S_{n}$ , in (2) below, may be interpreted as a gain from portfolio of $n$ European call options with the same strike price $r_{n}$ .

In this note we prove the following:

Theorem 1.

Let a sequence $0<r_{n}\to\infty$ and $X_{1},X_{2},...,X_{n},...$ be a sequence of non-negative independent identically distributed random variables. Then

[TABLE]

if and only if either

[TABLE]

or

[TABLE]

where $\nu$ is an exponential probability distribution with a probability density $f(x):=\lambda e^{-\lambda x}\,1_{[0,\infty)}(x)\,,\ (\lambda>0)$ .

In terms of Laplace transforms we have either $\mathfrak{L}([S;t]=\exp(-ct)$ or $\mathfrak{L}[S;t]=\exp a[\frac{\lambda}{\lambda+t}-1]$ , for $t\geq 0$ .

In (2) $"\Rightarrow"$ denotes a weak convergence of distributions and in (4) $"\stackrel{{\scriptstyle d}}{{=}}"$ means equality of probability distributions. A measure in (4) is a particular compound Poisson measure $e(m)$ , where $m:=a\nu$ .

2. Laplace transforms and the sufficiency part of Theorem 1.

For study of the weak convergence of non-negative random variables it is convenient to use their Laplace transforms. In that case, weak convergence of measures is equivalent to a point-wise convergence of their Laplace transforms; for instance cf. Feller (1966), Theorem 2a, p. 410.

Note that (1) and a simple calculation give a (cumulative) probability distribution function of $U_{r}(X)$ :

[TABLE]

Note that at zero there is jump of a size $F(r)$ .

Recall that the Laplace transform of a random variable $Y$ is defined as

$\mathfrak{L}[Y;t]:=\mathbb{E}[e^{-tY}],\ \mbox{for }\ t\geq 0.$ Hence

[TABLE]

This is so, because by (1) and (5) we get

[TABLE]

[Note that the function under last integral sign vanishes at zero].

Hence, from (6), for $r_{n}>0$ and independent identically distributed variables $(X_{k}),k=1,2,...,n$ with a distribution function $F$ , we have

[TABLE]

Thus $(G_{n})$ are continuous distribution function of finite measures on $[0,\infty)$ .

Here are examples of normalizing sequences $(r_{n})$ and random variables $(X_{n})$ such that $S_{n}\Rightarrow S$ with limits $S$ described in Theorem 1 by formulas (4) and (3), respectively.

Example 1. Let $(X_{n})$ be i.i.d. exponentially distributed with probability density $\lambda e^{-\lambda x}1_{[0,\infty)}(x)$ and for given $a>0$ , let us choose $r_{n}>0$ such that $e^{-\lambda r_{n}}=a/n,$ for all n such that $a/n<1$ . Then by (7), the limit of corresponding distributions $(G_{n})$ is as follows

[TABLE]

Finally, we have the Laplace transform of a limit $S$ :

[TABLE]

Here we recognize that $\exp[\frac{\lambda}{\lambda+t}-1]$ is a Laplace transform of compound Poisson variable $\sum_{k=0}^{N_{1}}\,Y_{k}$ , where $(Y_{n})$ are i.i.d. exponentially distributed with parameter $\lambda$ and independent of a Poisson variable $N_{1}$ .

Example 2. Let random variables $(X_{n})$ be independent with the same (cumulative) distribution function $F$ which has the probability density

$(\frac{2}{\pi})^{1/2}\exp(-x^{2}/2)1_{(0,\infty)}(x)$ . Note that a function $k(x):=(\frac{2}{\pi})^{1/2}x^{-2}e^{-x^{2}/2}$ on $(0,\infty)$ is continuous and strictly decreasing from infinity to zero. Thus for a given $c>0$ we may choose $r_{n}$ as the solution of the equation $k(x)=c/n$ , that is, $n\,(\frac{2}{\pi})^{1/2}(r_{n}^{2}\,e^{r_{n}^{2}/2})^{-1}=c$ .

Taking into account the above, from (7), $G(0):=\lim_{n\to\infty}G_{n}(0)=0$ and for $y>0$ we have

[TABLE]

All in all, in this example, $G(dx)=c\delta_{0}(dx)$ is a distribution function of a measure concentrated at zero with a mass $c$ . Consequently,

[TABLE]

or in other words, the limit $S=c$ with P.1.

This completes a proof of the necessity part of Theorem 1.

3. Some functional equations and their solutions.

From now on we assume that $S_{n}\Rightarrow S$ which (by (7)) is equivalent to

[TABLE]

where $G_{n}$ are distribution functions of finite measures on the positive half-line.

Lemma 1.

(i) If $S_{n}\Rightarrow S$ , in (2), then distribution functions $(G_{n})$ are uniformly bounded, that is, $\sup_{(n\geq 1)}G_{n}(+\infty)\leq K<\infty$ .

(ii) If $G$ is a limit point of $\{G_{n}\}$ then $G(\{\infty\})=0$ .

Proof.

(i) In contrary, assume that there exists sub-sequence $n_{k},\ k=1,2,...$ such that $\lim_{k\to\infty}G_{n_{k}}(+\infty)=\infty$ then from (6) we get that

[TABLE]

which contradict the fact that Laplace transform are positive functions.

Hence distribution functions $(G_{n})$ of finite measures on the positive half-line and uniformly bounded (conditionally compact).

(ii) Let $G$ on $[0,\infty]$ be a weak limit of subsequence of distribution functions $(G_{n})$ . Then

[TABLE]

Taking $t\to 0$ , the function under integral sign tends to zero and hence we must have $1=\mathfrak{L}(S;0)=\exp(-G(\{\infty\})$ which means that $G(\{\infty\})=0$ . This completes a proof of Lemma 1. ∎

Remark 1.

It may be worthy to recall that a family of probability measures on compact topological space is compact convolution semigroup, in weak topology; cf. Parthasarathy(1967), Theorem 6.4. p. 45.**

Remark 2.

For a discussion below recall that for (probability) distribution functions $F_{n}$ and $F$ we have*:*

(a) if $F_{n}\Rightarrow F,\ a_{n}\to a$ , $a$ is continuity point of $F$ , then $F_{n}(a_{n})\to F(a)$ ;**

(b) if $F_{n}$ converges to $F$ point wise and $F$ is continuous then $F_{n}\to F$ uniformly.**

From Lemma 1 and the limit in (7) we have that $G_{n}\Rightarrow G$ (weak convergence) to a distribution function of finite (not necessarily probability) measure $G$ . To identify $G$ we introduce auxiliary functions:

[TABLE]

In particular from Remark 2, we have that if $u$ is a continuity point of $H$ (note that $G$ may have a jump at $u$ ) and $u_{n}\to u$ then $H_{n}(u_{n})\to H(u)$ .

From (7), for $1\leq u<\infty$ , we have that

[TABLE]

Lemma 2.

Assume that $w_{n}:=r_{n+1}-r_{n}\to w\in[0,\infty]$ ( or choose a sub-sequence). Then

[TABLE]

Proof.

Since $r_{n+1}=w_{n}+r_{n}$ , from (10), we infere that

[TABLE]

∎

What are possible functions $H$ satisfying a functional equation (11)? We consider the three possible cases for a limit $w$ .

Lemma 3.

If $w_{n}:=r_{n+1}-r_{n}\to w\in(0,\infty]$ (or choose a subsequence) then $H(u)=0$ and $H(u)=H(\infty)$ are the only solutions to a functional equation $H(u)+H(1+w)=H(u+w)$ , where $0<u<\infty$ .

Proof.

A case $w=\infty$ is obvious as $H(\infty)$ is finite so we get $H(u)=0$ .

Let $0<w<\infty$ . Then by the induction argument we have

[TABLE]

Since left hand side has a finite limit $H(\infty)$ as $k\to\infty$ therefore $H(1+w)=0$ . Consequently, $H(u+kw)=H(u),k\geq 1$ which gives $H(u)=H(\infty)=const$ and completes a proof. ∎

For the reaming case $w=0$ (i.e., $\lim_{n\to\infty}w_{n}=0$ ; cf. Lemma 2) we need the following elementary fact.

Fact. If $0<r_{n}\nearrow\infty$ and $r_{n+1}-r_{n}\to 0$ as $n\to\infty$ the for each number $0<c<1$ there exists a sub-sequence $(k_{n})$ such that $r_{k_{n}}-r_{n}\to c$ .

Proof.

Choose explicitly $k_{n}:=\sup\{k\geq n:r_{k}-r_{n}<c\}$ . Then we have that $r_{k_{n}}-r_{n}<c\leq r_{k_{n}+1}-r_{n}$ and a length of an interval containing $c$ is $r_{k_{n}+1}-r_{k_{n}}\to 0$ as $n\to\infty$ , which completes an argument. ∎

Lemma 4.

If $r_{n+1}-r_{n}\to 0$ then for any number $0<a<1$ $a$ there exists $b\geq 1$ (depending on $a$ ) such that

[TABLE]

Proof.

From previous Fact, there exist a sub- sequence $(k_{n})$ such that $v_{n}:=r_{k_{n}}-r_{n}\to a$ . Furthermore,

[TABLE]

Since limits in square brackets [ …] exist and are $H(u+a)$ and $H(1+a)$ , respectively, we infer that $b:=\lim_{n\to\infty}\frac{k_{n}}{n}$ exists. Finally, we get a formula $H(u)=b\,H(u+a)-H(1+a)$ , which completes a proof of the lemma. ∎

Remark 3.

Note that for $b=1$ the equation (10) coincides with the one given in Lemma 3.

To solve the functional equation (10) we need some facts that are quoted in the Appendix and from them we have:

Lemma 5.

The only solutions to an equation (10) in Lemma 4 are

[TABLE]

Proof.

Using Appendix for

$f(u):=H(u),\psi(y):=H(1+y),\ \mbox{and$ \phi $such that}\ \phi(a):=b^{-1},$

we retrieve our equation (11). Thus from (12), $H(u)=const$ if $f_{1}(x)=0$ .

In a case $f_{1}(x)=\alpha e^{\gamma x}$ we get $e^{\gamma a}=b^{-1}<1$ which implies $\gamma<0$ and that gives

[TABLE]

From first restriction we get $H(u)=\alpha(e^{\gamma u}-e^{\gamma})$ . Thus the second restriction imposes that $\alpha<0$ which completes arguments for (13). ∎

4. Proof of the necessity part of Theorem 1.

From Lemma 2, we have that $G(y)-G(1)=\int_{1}^{y}(1-e^{-x})dH(x),y>0.$ and from Lemma 5 we know forms of $H$ ( introduced in Lemma 2). Consequently, we have that either $G(y)=c>0$ (constant) for $y>0$ and $G(0)=0$ or

[TABLE]

This may be a written as

[TABLE]

Using (7) we have

[TABLE]

which completes a proof of Theorem 1.

5. Appendix.

For ease of reference let us quote the following fact.

Lemma 6.

Let functions $f$ , $\phi$ and $\psi$ be defined on real (or positive) numbers and satisfy equations

[TABLE]

If $f$ is continuous (or bounded on a set of positive Lebesque measure) then solutions to equation (14), part (i), are

[TABLE]

and for part (ii) solutions are

[TABLE]

cf. Aczel (1962), Theorem 8 on p. 22 or Aczel (1966), Chapter 3, Theorem 1 on p.150.

6. A historical note. A problem of the characterization of all possible limit distributions for sequences $U_{r_{n}}(X_{1})+U_{r_{n}}(X_{2})+...+U_{r_{n}}(X_{n})$ for positive variables was posed by Kazimierz Urbanik around 1971. (For many years the notation $T_{r}$ instead of $U_{r}$ was used.) Proofs presented in this paper are taken from MS Thesis [5] written in Polish language in 1972 and they were never publish before in English. References [3] and [7]-[9] are added to this article to point out the current state of the problem.

On the other hand, the Reader should be aware that the definition of shrinking s-transforms $U_{r}$ have a natural generalization to vector valued random variables $X$ as follows: $U_{r}(0):=0$ and

[TABLE]

Urbanik problem for a Hilbert space valued, not necessarily identically distributed, random vectors was completely solved in 1977 where in the proof the Choquet Theorem on extreme points in convex compact sets was used; cf. Jurek (1981).

However, knowing solutions for real random variables it is not obvious how to infer characterizations for positive variables. Furthermore, Hilbert space generality may not be accessible for a wider audience. Those concerns led us to the decision to present here elementary proofs for non-negative random variables from [5].

Let us also add here that more recently, in Bradley and Jurek (2015), stochastic independence of variables in Theorem 1 was replaced by the weak dependence (strong mixing).

Last but not least, potential applications in a mathematical finance (mentioned in the Introduction) of which the Author was not aware 50 years ago, may generate some additional interest for this quite straightforward and elementary proofs.

Acknowledgments. Reviewer’s comments lead to a better presentation of some arguments and improved the language of this note.

References.

[1] J. Aczel (1962), On applications and theory of functional equations,

Birkhauser Verlag, Basel 1969.

[2] J. Aczel (1966), Lectures on functional equations and their

applications, Academic Press, New York.

[3] R. C. Bradley and Z.J. Jurek (2015), On a central limit theorem

for shrunken weakly dependent random variables, Houston Journal of

Mathematics, vol. 41, No. 2, pp. 621-638.

[4] W. Feller(1966), An Introduction to Probability Theory, vol. 2,

J. Wiley $\&$ Sons, New York 1966.

[5] Z. J. Jurek (1972), On a class of limit distributions, University of

Wroclaw, MS Thesis (in Polish), University Archives, call number:

W IV-5200/Jurek Zbigniew; (To Archives e-mail address: [email protected])

[6] Z. J. Jurek (1981), Limit distributions for sums of shrunken random

variables, Dissertationes Mathematicae,vol. CLXXXV, Polskie

Wydawnictwo Naukowe, Warszawa, 1981, pp. 50;

(accepted for a publication on November 29, 1977).

[7] Z. J. Jurek and J. D. Mason (1993), Operator- limit distributions in

probability theory, J. Wiley $\&$ Sons, Inc. New York

[8] M. M. Meerscheart and H-P Schefller (2001), Limit distributions for

sums of independent random vectors, J. Wiley $\&$ Sons, Inc. New York.

[9] K. R. Parthasarathy (1967), Probability measures on metric spaces,

Academic Press, 1967.

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On a limit theorem for a non-linear scaling

Theorem 1**.**

Lemma 1**.**

Proof.

Remark 1**.**

Remark 2**.**

Lemma 2**.**

Proof.

Lemma 3**.**

Proof.

Proof.

Lemma 4**.**

Proof.

Remark 3**.**

Lemma 5**.**

Proof.

Lemma 6**.**

Theorem 1.

Lemma 1.

Remark 1.

Remark 2.

Lemma 2.

Lemma 3.

Lemma 4.

Remark 3.

Lemma 5.

Lemma 6.