Random Dimension Low Sample Size Asymptotics

Abstract

A first investigation of high-dimensional low-sample-size (HDLSS) asymptotics, Hall, Marron and Neeman (2005) discovered a surprisingly rigid geometric structure. A sample of size k taken from the standard m-dimensional normal distribution is for large m close to the vertices of the k-dimensional simplex in m-dimensional vector space. It follows from the analysis of three geometric statistics: the length of an observation, the distance between any two independent observations and the angle between these vectors. We generalize and refine the results constructing the second order Chebyshev-Edgeworth expansions under assumption that the data dimension is random and different scaling factors are chosen.

1 Three Geometric Statistics of Gaussian Vectors

We continue to study properties of high-dimensional Gaussian random vectors. In our earlier papers Christoph, Prokhorov and Ulyanov [8] and Bobkov, Naumov and Ulyanov [5] two-sided bounds were constructed for a probability density function of the distance of a Gaussian random element Y with zero mean from a point a in a Hilbert space $\mathbb {H}$. We get new results for basic geometric statistics connected with high-dimensional random normal vectors.

Let $\mathbf {X}_1=(X_{1,1},...,X_{1,m})^T$,..., $\mathbf {X}_k=(X_{k,1}...,X_{k,m})^T$ be a random sample.

In a high-dimension low-sample-size (HDLSS) data it is assumed that dimension m tends to infinity and sample size k is fixed.

One of the first investigation of HDLSS data was done in Hall, Marron and Neeman (2005) [14]. It became the basis of research in high-dimensional mathematical statistics. See a recent survey on HDLSS asymptotics and its applications in Aoshima et al. [1]. Further development see e.g. in Fujikoshi, Ulyanov and Shimizu [12] when both m and k may tend to infinity. This is an important framework of the current data analysis called Big data. In [14] it was discovered a surprisingly rigid geometric structure. A sample of size k taken from the standard m-dimensional normal distribution is close for large m to the vertices of the k-dimensional simplex in $\mathbb {R}^m$. It follows from the analysis of three geometric statistics:

• the length $||\mathbf {X}_i||_m$ of an observation,

• the distance $||\mathbf {X}_i- \mathbf {X}_j||_m$ between any two independent observations,

• and the angle $\theta _m = \mathrm {ang}(\mathbf {X}_i,\mathbf {X}_j)$ between these vectors.

We generalize and refine the results constructing the second order Chebyshev-Edgeworth expansions under assumption that the data dimension is random and different scaling factors are chosen.

In case of $\dim \mathbb {H} < \infty$ we consider a sample of size k when the dimension of the observations is a random variable $N_n$ with values in $\mathbb {N}_+ = \{1, 2, \ldots \}$.

The present work continues our investigations in Christoph and Ulyanov [9] on these three geometric statistics of Gaussian vectors with randomly distributed dimension $N_n$ which depends on parameter $n \in \mathbb {N}_+$ and $N_n \rightarrow \infty$ in probability as $n \rightarrow \infty$. Let the vectors $\mathbf {X}_1, ..., \mathbf {X}_k$ and $N_1,N_2, ...$ be defined on one and the same probability space and it is assumed that they are independent. If $T_m := T_m \left( \mathbf {X}_1,..., \mathbf {X}_k\right)$ is some statistic of the vectors $\mathbf {X}_1,..., \mathbf {X}_k$ with non-random dimension $m \in \mathbb {N}_+$ then the random variable $T_{N_n}=T_{N_n}(\omega )$ is defined as:

$$ T_{N_n} (\omega ) := T_{N_n(\omega )} \left( \mathbf {X}_1(\omega ),..., \mathbf {X}_k (\omega )\right) ,\quad \omega \in \varOmega \quad \text{ and } \quad n \in \mathbb {N}_+. $$

Therefore, the statistics $T_{N_n}$ based on statistics $T_m$ are constructed from the sample $\{\mathbf {X}_1,..., \mathbf {X}_k\}$, where these vectors have the dimension $N_n$.

In [9], the distribution function of the normalized angle $\theta _m =\mathrm {ang}(\mathbf {X}_i,\mathbf {X}_j)$ was approximated by a second order Chebyshev-Edgeworth expansion with a bound $\le C m^{-2}$ for all $m \in \mathbb {N}_+$. Furthermore, the fixed dimension m of the Gaussian vectors was substituted by a random number $N_n$ and expansions for statistics $\theta _{N_n}$ were proved.

A natural question arises whether similar results hold for the length $||\mathbf {X}_i||_{N_n}$ and the distance $||\mathbf {X}_i- \mathbf {X}_j||_{N_n}$ of Gaussian vectors with random dimension ${N_n}$.

Two cases of random dimensions (or random sample sizes) $N_n$ are considered as e.g. in Bening, Galieva and Korolev [2], Christoph, Monakhov and Ulyanov [7] and Christoph and Ulyanov [9]:

• i) The random dimension $N_n=N_n(r) \in \mathbb {N}_+$ has negative binomial distribution displaced by 1 with probability of success 1/n, positive parameter $r>0$ and probabilities

  $$\begin{aligned} \mathbb {P}(N_n(r)=j)= \frac{\varGamma (j+r-1)}{\varGamma (j) \, \varGamma (r)}\left( \frac{1}{n} \right) ^{r}\left( 1 - \frac{1}{n} \right) ^{j-1}, \,\,\, j \in \mathbb {N}_+ . \end{aligned}$$

  (1)

• ii) The random dimension $N_n=N_n(s) \in \mathbb {N}_+$ is discrete Pareto-like distributed with parameters $n \in \mathbb {N}_+$, $s>0$ and distribution function

  $$\begin{aligned} \mathbb {P}(N_n(s)\le k)=\left( \frac{k}{s+k}\right) ^{n}\quad \text{ where } \quad N_n(s)= \max _{1\le j \le n} Y_j(s), \end{aligned}$$

  (2)

  and $Y(s), Y_1(s), Y_2(s), . . .$, are independent discrete Pareto II distributed random variables with the common distribution

  $$\begin{aligned} \mathbb {P}\big (Y(s)\le k\big ) =\frac{k}{s+k} \quad \text{ and } \quad \mathbb {P}(Y(s)=k)=\frac{s}{(s+k)\,(s+k-1)}, \,\,k \in \mathbb {N_+}. \end{aligned}$$

  (3)

  The discrete Y(s) on integers is the discretized continuous Pareto II (Lomax) random variable, see Buddana and Kozubowski [6].

Both cases of random dimensions of the Gaussian vectors are also interesting because $\mathbb {E}N_n(r) = r (n-1) +1 < \infty$ and $\mathbb {E}N_n(s) = \infty$, which has an influence on the normalization factors.

The rest of the paper is organized as follows: In Sect. 2, Chebyshev-Edgeworth expansions are proved for the geometric statistics of Gaussian vectors with fixed dimension m. Section 3 presents the transfer theorem for results with fixed sample size (in our case the dimension of the vectors) m to those with random sample size $N_n$. The main results are given in Sects. 4 and 5 when the random sample size is negative binomial $N_n(r)$ or discrete Pareto-like $N_n(s)$ distributed, respectively. In Sect. 6 the main results are proved.

2 Approximation for Geometric Statistics of m-Dimensional Normal Vectors

Let $\mathbf {X}_i=(X_{i,1},...,X_{i,m})^T$,..., $\mathbf {X}_j=(X_{j,1}...,X_{j,m})^T$ be m-dimensional vectors chosen from a sample $\{\mathbf {X}_1, .... , \mathbf {X}_k\}$ of normal distribution $\mathcal {N}(\mathbf {0}_m, {\mathrm{I}}_m)$ with mean vectors $\mathbb {E}\mathbf {X}_k = \mathbf {0}_m$ and covariance matrix $\mathrm{I}_m$ for $1 \le i<j \le k\le m$.

The length of the vector $\mathbf {X}_j$ is defined by the Euclidean distance $|| \cdot ||_m$:

$$\begin{aligned} ||\mathbf {X}_i||_m = S_m^{1/2} \quad \text{ with } \quad S_m = \sum \nolimits _{k=1}^m X_{i,k}^2\,. \end{aligned}$$

(4)

and similarly the distance $||\mathbf {X}_i- \mathbf {X}_j||_m$ between any two independent vectors

$$\begin{aligned} ||\mathbf {X}_i - \mathbf {X}_i||_m = \sum \nolimits _{k=1}^m \left( X_{i,k} - X_{j,k}\right) ^2\,. \end{aligned}$$

(5)

The distribution of distance $||\mathbf {X}_i- \mathbf {X}_j||_m$ is closely linked to the distribution of length $||\mathbf {X}_i||_m$, since $(X_{i,k} - X_{j,k})/\sqrt{2}$ has also standard normal distribution $\varPhi (x)$. Therefore

$$\begin{aligned} \mathbb {P}(||\mathbf {X}_i- \mathbf {X}_j||_m/\sqrt{2} \le x) = \mathbb {P}(||\mathbf {X}_i||_m \le x). \end{aligned}$$

(6)

The angle $\theta _m = \mathrm {ang}(\mathbf {X}_i,\mathbf {X}_j)$ between these two independent vectors with vertex at the origin and the sample correlation coefficient $R_m(\mathbf {X}_i, \mathbf {X}_j)$ are connected by:

$$\begin{aligned} \cos \theta _m = \frac{||\mathbf {X}_i||_m^2+||\mathbf {X}_j||_m^2 - ||\mathbf {X}_i -\mathbf {X}_j||_m^2}{2\,||\mathbf {X}_i||_m\,\,||\mathbf {X}_j||_m} =R_m(\mathbf {X}_i,\mathbf {X}_j)=R_m. \end{aligned}$$

(7)

Hall, Marron and Neeman [14] showed

• for the length $||\mathbf {X}_i||_m = \sqrt{m} +\mathcal {O}_p(1)$,

• for the distance $||\mathbf {X}_i- \mathbf {X}_j||_m =\sqrt{2m} +\mathcal {O}_p(1)$ with $i \not = j$ and

• for the $\theta _m$ = angle $\mathrm {ang}(\mathbf {X}_i,\mathbf {X}_j) = \frac{1}{2}\pi + \mathcal {O}_p(m^{-1/2})$ with $i \not = j$,

where $1 \le i<j\le k \le m$ and $\mathcal {O}_p$ refers to the stochastic boundedness.

The length of the vector $\mathbf {X}_i$ drawn from an m-dimensional normal distribution $\mathcal {N}({\mathbf {0},\mathrm{I}}_m)$ is defined in (4) as $||\mathbf {X}_i||_m = S_m^{1/2}$, where the statistics $S_m$ as a sum of the squares of m independent standard normal random variables has chi-square distribution with m degrees of freedom and

$$\begin{aligned} V_m = \frac{S_m -m}{\sqrt{2\,m} } \end{aligned}$$

(8)

is asymptotically standard normally distributed. With the two-term Chebyshev-Edgeworth expansions in the central limit theorem for the distribution function of $V_m$, the following inequality results for all $m \in \mathbb {N}$

$$ \left| P\Big (V_m \le x\Big ) - \varPhi (x) - \varphi (x)\,\Big (\frac{\lambda _3\,H_2(x)}{6\,\sqrt{m}} + \frac{\lambda _3^2 \,H_5(x)}{72\,m} + \frac{\lambda _4\,H_3(x)}{24\,m}\Big )\right| \le \frac{C}{m^{3/2}}$$

where $H_2(x) =x^2 -1$, $H_3(x)=x^3 - 3 x$, $H_5(x)= x^5 - 10x^3 + 15 x$ are the Chebyshev-Hermite polynomials, skewness $\lambda _3 = \sqrt{8}$ and excess kurtosis $\lambda _4 = 12$ of $S_1$, see Petrov [19, Sec. 5.7, Theorem 5.18].

Then $S_m =m(1+ \sqrt{2/m}\,V_m)$ and Tayor expansion of $(1+u)^{1/2}$ lead to

$$\begin{aligned} ||\mathbf {X}_i||_m = S_m^{1/2} = \sqrt{m} \left( 1+ \frac{1}{\sqrt{2\,m}}\,V_m - \frac{1}{4\,m}\,V^2_m + \frac{\sqrt{2}}{8\,m^{3/2}}V^3_m + ...\right) \end{aligned}$$

(9)

Define the statistics

$$\begin{aligned} Z_m = \sqrt{2}\Big (\frac{\displaystyle ||\mathbf {X}_i||_m}{\displaystyle \sqrt{m}}\,- 1\Big ) \quad \text{ and } \quad Z_m^* = \sqrt{2}\Big (\frac{\displaystyle ||\mathbf {X}_i - \mathbf {X}_j||_m}{\displaystyle \sqrt{2\,m}}\,- 1\Big ), \end{aligned}$$

(10)

then (6) results in

$$\begin{aligned} P\Big (\sqrt{m}\,Z_m \le x\Big ) = P\Big (\sqrt{m}\,Z_m^* \le x\Big ). \end{aligned}$$

(11)

It follows from (9) that the statistic $T_1=\sqrt{m} Z_m$ holds

$$\begin{aligned} T_1=\sqrt{m} Z_m = V_m- \frac{\sqrt{2}}{4 \sqrt{m}} \,V_m^2 + \frac{\sqrt{1}}{4 \,m} \,V_m^3 + ... \end{aligned}$$

(12)

Following the sketch of the proof in Kawaguchi, Ulyanov and Fujikoshi [16, Theorem 1] (The coefficients in the polynomial $l_2(x)$ are incorrect.) and calculating the characteristic function $f_{T_1}(t)$, we obtain

$$\begin{aligned} f_{T_1}(t)= & {} \mathbb {E}\left[ {\mathrm{e}}^{{\mathrm{i}} t V_m}\,\left( 1 - \frac{\sqrt{2}({\mathrm{i}} t)}{4 \sqrt{m}} V_m^2 + \frac{({\mathrm{i}} t)}{4\,m} V_m^3 + \frac{({\mathrm{i}} t)^2}{16\,m} V_m^4 + \mathcal {O}_p(m^{-3/2})\right) \right] \nonumber \\&= {\mathrm{e}}^{-t^2/2}\left( 1 - \frac{\sqrt{2}(({\mathrm{i}} t)^3 + 3 ({\mathrm{i}} t))}{12 \sqrt{m}} + \frac{({\mathrm{i}} t)^6 - 6 ({\mathrm{i}} t)^4 - 9({\mathrm{i}} t)^2}{144\,m} )\right) + \mathcal {O}(m^{-3/2}). \end{aligned}$$

(13)

This results in the related expansion of the corresponding distribution function:

Proposition 1

Let $\mathbf {X}_i$ be a vector drawn from an m-dimensional normal distribution $\mathcal {N}({\mathbf {0}_m,\mathrm{I}}_m)$. Then with the asymptotic expansion for the distribution of normalized length $Z_m = \sqrt{2}\Big (\frac{\displaystyle ||\mathbf {X}_i||_m}{\displaystyle \sqrt{m}}\,- 1\Big )$ we obtain the following inequality for all $m \in \mathbb {N}$:

$$\begin{aligned} \left| P\Big (\sqrt{m}\,Z_m \le x\Big ) - \varPhi (x) - \varphi (x)\,\Big (\frac{x^2 - 4}{6 \, \sqrt{2\,m}}\, + \frac{x^5 -16 x^3+ 24 x}{144\,m}\Big )\right| \le \frac{C}{m^{3/2}}. \end{aligned}$$

(14)

Corollary 1

Let $\mathbf {X}_i$ and $\mathbf {X}_j$, $i \not = j$ be independent random vectors with an m-dimensional normal distribution $\mathcal {N}({\mathbf {0}_m,\mathrm{I}}_m)$. Due to (11), distribution function of the normalized distance $Z_m^* = \sqrt{2}\Big (\frac{\displaystyle ||\mathbf {X}_i - \mathbf {X}_j||_m}{\displaystyle \sqrt{2\,m}}\,- 1\Big )$ has the same asymptotic expansion as the distribution of normalized length $Z_m$ and inequality (14) with replacing $Z_m$ by $Z_m^*$.

Second order Chebyshev-Edgeworth expansion of the angle $\theta _m = \mathrm {ang}(\mathbf {X}_i,\mathbf {X}_j)$ between independent vectors $\mathbf {X}_i$ and $\mathbf {X}_j$ with vertex at the origin and the corresponding sample correlation coefficient $R_m(\mathbf {X}_i, \mathbf {X}_j)$ with computable error bounds of approximation are shown in Christoph and Ulyanov [9, Section 2], using results of Konishi [17, Sect. 4], Johnson, Kotz and Balakrishnan [15, Chap. 32], Christoph, Ulyanov and Fujikoshi [11]:

$$\begin{aligned} \sup \nolimits _x \left| P\Big (\sqrt{m}\,\,R_m \le x\Big ) \,-\, \varPhi (x) - \frac{x^3 - 5 x}{4\, m} \,\varphi (x) \right| \le \frac{B_1}{m^2} \end{aligned}$$

(15)

and

$$\begin{aligned} \sup \limits \nolimits _{x} \Big | P\left( \sqrt{m}(\theta _m - \frac{\pi }{2}) \le x\right) - \varPhi (x) - \frac{x^3 -15 x}{12 \,m} \,\varphi (x) \Big |\le \frac{B_2}{m^2}. \end{aligned}$$

(16)

The estimates (15) and (16) were used in Christoph and Ulyanov [9] to obtain second order approximations the statistics $R_{N_n}$ and $\varTheta _{N_n}= \theta _{N_n} - \pi /2$ when the non-random dimension m of the vectors is replaced be a random dimension $N_n$, where the random dimension $N_n \rightarrow \infty$ in probability when the parameter $n \rightarrow \infty$.

Analogous results for the statistics $||\mathbf {X}_i||_m$ and $||\mathbf {X}_i- \mathbf {X}_j||_m$ are proven in Sects. 4 and 5 below, when the non-random dimension m is replaced be a random dimension $N_n$.

3 Auxiliary Proposition

In this section, expansions for the distribution function of statistics $T_{N_n}$ obtained from samples with random sample size (here with random dimension $N_n$ of the considered vectors $\mathbf {X}_i$) are obtained. These depend directly on the expansions concerning statistics $T_m$ based on non-random samples size m and expansions regarding the random sample size $N_n$.

First we formulate the conditions determining expansions for the statistic $T_m$ with $\mathbb {E} T_m =0$ and the normalized random dimension $N_n$:

Assumption A:

Given $\gamma \in \{- 1/2, 0, 1/2\}$, $a > 1$, $C_1 > 0$ and differentiable functions $f_1(x), f_2(x)$ with bounded derivatives $f'_1(x), f'_2(x)$ such that

$$\begin{aligned} \sup \nolimits _{x}\Big |\mathbb {P}\big (m^{\gamma } T_m \le x \big ) - \varPhi (x) - \frac{f_1(x)}{\sqrt{m}} - \frac{f_2(x)}{m} \Big | \le \frac{C_1}{m^{a}}\quad \text{ for } \text{ all } \quad m \in \mathbb {N}. \end{aligned}$$

(17)

Remark 1

Statistics satisfying Assumption A are shown in (14), (15) and (16).

Assumption B:

Given constants $b > 0$ and $C_2 > 0$, real numbers $g_n$ with $0<g_n \uparrow \infty$ if $n \rightarrow \infty$, a distribution function H(y) with $H(0+) = 0$ and a function $h_2(y)$of bounded variation that

$$\begin{aligned} \sup \limits _{y \ge 0} \left| \mathbb {P}\left( \frac{N_n}{g_n} \le y\right) - H(y) - \frac{h_2(y) \, \mathbb {I}_{\{b>1\}}(b)}{n} \right| \le \frac{C_2}{n^{b}} \quad \text{ for } \text{ all } \quad n \ge 1. \end{aligned}$$

(18)

where $\mathbb {I}_{A}(x)= \left\{ \begin{array}{ll} 1,\quad &{} x\in A \\ 0, &{} x \notin A \end{array}\right.$    defines the indicator function of a set $A \subset \mathbb {R}$.

Remark 2

The random dimensions $N_n(r)$ and $N_n(s)$ given in (1) and (2), respectively, fulfill Assumption B as shown in [9, Propositions 1 and 2], see (29) and (39) below.

Proposition 2

Let $\gamma \in \{1/2,\, 0,\, -1/2\}$ and both Assumption A and B as well as the following requirements on H(.) and $h_2(.)$ are fulfilled

$$\begin{aligned} \left. \begin{array}{llll} i: \quad &{} H(1/g_n) \le c_1\,g_n^{- b} \qquad &{} \text{ for } &{} b> 0,\\ ii: \quad &{} \int \nolimits _0^{1/g_n} y^{-\,1/2} dH(y) \le c_2\,g_n^{- b + 1/2} \qquad &{} \text{ for } &{} b> 1/2,\\ iii: \quad &{} \int \nolimits _0^{1/g_n} y^{-\,1} dH(y)\le c_3\,g_n^{- b + 1} \qquad &{} \text{ for } &{} b>1, \end{array}\,\right\} \end{aligned}$$

(19)

$$\begin{aligned} \left. \begin{array}{llll} i: &{} h_2(0) = 0, \quad \text{ and } \quad |h_2(1/g_n)| \le c_4\,n\,g_n^{- b}&{} \text{ for } &{} b>1,\\ ii: &{} \int \nolimits _0^{1/g_n} y^{-\,1} |h_2(y)| dy \le c_5\,n \,g_n^{- b}&{} \text{ for } &{} b>1, \end{array}\,\right\} \end{aligned}$$

(20)

where b is the convergence rate in (18). Then for all $n \ge 1$ is valid:

$$\begin{aligned} \sup \limits _{x \in \mathbb {R}}\Big | \mathbb {P}\Big (g_n^{\gamma } T_{N_n} \le x\Big ) - G_{n,2}(x)\Big | \le C_1 \,\mathbb {E} \left( N_n^{- a}\right) + (C_3 D_n + C_4)\, n^{- b}+ I_n , \end{aligned}$$

(21)

where with $a>1, b>0, f_1(z), f_2(z), h_{2}(y)$ are given in (17) and (18)

$$\begin{aligned} G_{n,2}(x) = \left\{ \begin{array}{lr} \int \limits ^{\infty }_{0} \varPhi (x\,y^\gamma ) \mathrm{d}H(y), &{} 0<b \le 1/2, \\ \int \limits _{0}^{\infty } \Big (\!\varPhi (xy^\gamma ) + \frac{\displaystyle f_1(x\,y^\gamma )}{\displaystyle \sqrt{g_n y}} \Big ) \mathrm{d}H(y) =: G_{n,1}(x), &{} 1/2 < b \le 1,\\ G_{n,1}(x) + \int \limits ^{\infty }_{0} \frac{\displaystyle f_2(x\,y^\gamma )}{\displaystyle g_n y}\mathrm{d}H(y) + \int \limits _{0}^{\infty } \frac{\displaystyle \varPhi (x\,y^\gamma )}{\displaystyle n} \mathrm{d}h_2(y), &{} b > 1,\end{array} \right\} \end{aligned}$$

(22)

$$\begin{aligned} D_n= \sup _x \int _{1/g_n}^\infty \left| \frac{\displaystyle \partial }{\displaystyle \partial y} \left( \varPhi (xy^{\gamma }) + \frac{f_1(xy^{\gamma })}{\sqrt{g_n y}} + \frac{f_2(xy^{\gamma })}{y g_n} \right) \right| \mathrm{d}y, \end{aligned}$$

(23)

$$\begin{aligned} I_n = \sup \nolimits _x \left( |I_1(x,n)| + |I_2(x,n)|\right) , \end{aligned}$$

(24)

$$\begin{aligned} I_1(x,n) = \int \nolimits _{1/g_n}^\infty \Big (\frac{f_1(x\,y^\gamma )\,\,{\mathbb {I}}_{(0 , 1/2]}(b)}{\sqrt{g_n y\,}} + \frac{f_2(x\,y^\gamma )}{g_n\,y}\Big ) \mathrm{d}H(y), \quad \quad b\le 1, \end{aligned}$$

(25)

and

$$\begin{aligned} I_2(x,n) = \int \nolimits _{1/g_n}^\infty \Big (\frac{f_1(x\,y^\gamma )}{n\,\sqrt{g_n y\,}} + \frac{f_2(x\,y^\gamma )}{n\,g_n y}\Big ) \mathrm{d}h_2(y), \quad \quad b > 1. \end{aligned}$$

(26)

The constants $C_1, C_3, C_4$ are independent of n.

Proof

The proof is based on the statement in [2, Theorem 3.1] for $\gamma \ge 0$. Since in Theorems 1 and 2 in the present paper as well as in Christoph and Ulyanov [9, Theorems 1 and 2] the case $\gamma = - 1/2$ is also considered, therefore the proof was adapted to $\gamma \in \{1/2,\, 0,\, -1/2\}$ in [9]. The conditions (19) and (20) guarantee integration range $(0 , \infty )$ of the integrals in (22). The approximation function $G_{n,2}(x)$ in (22) is now a polynomial in $g_n^{-1/2}$ and $n^{-1/2}$. Present Proposition 2 differs from Theorems 1 and 2 in [9] only by the term $f_1(xy^{\gamma })\,(g_n y)^{- 1/2}$ and the added condition (19ii) to estimate this term. Therefore here the details are omitted.    $\square$

Remark 3

The domain $[1/g_n, \infty )$ of integration depends on $g_n$ in (23), (25) and (26). Some of the integrals in (25) and (26) could tend to infinity with $1/g_n \rightarrow 0$ as $n \rightarrow \infty$ and thus worsen the convergence rates of the corresponding terms. See (47) in Sect. 6.

In the next two sections we consider the statistics $Z_m$ and $Z_m^*$ defined in (10) and the cases when the random dimension $N_n$ is given in either (1) or (2). We use Proposition 2 when the limit distributions of scaled statistics $Z_{N_n}$ are scale mixtures $G_\gamma (x) = \int _0^\infty \varPhi (x\, y^{\gamma }) \mathrm{d}H(y)$ with $\gamma \in \{1/2,\, 0,\, -1/2\}$ that can be expressed in terms of the well-known distributions. We obtain non-asymptotic results for the statistics $Z_{N_n}$ and $Z_{N_n}^*$, using second order approximations the statistics $Z_m$ and $Z_m^*$ given in (14) as well as for the random sample size $N_n$. In both cases the jumps of the distribution function of the random sample size $N_n$ only affect the function $h_2(y)$ in formula (18).

4 The Random Dimension $N_n(r)$ is Negative Binomial Distributed

The negative binomial distributed dimension $N_n(r)$ has probability mass function (1)) and $g_n = \mathbb {E}(N_n(r))= r\,(n-1) +1$. Schluter and Trede [21] (Sect. 2.1) underline the advantage of this distribution compared to the Poisson distribution for counting processes. They showed in a general unifying framework

$$\begin{aligned} \lim \nolimits _{n\rightarrow \infty } \sup \nolimits _y\left| \mathbb {P}(N_n(r)/g_n \le y) - G_{r,r}(y)\right| = 0, \end{aligned}$$

(27)

where $G_{r,r}(y)$ is the Gamma distribution function with the identical shape and scale parameters $r >0$ and density

$$\begin{aligned} g_{r,r}(y)=\frac{r^r}{\varGamma (r)}\,\ y^{r - 1} \, e^{- r y}\,\,{\mathbb {I}}_{(0\,,\,\infty )}(y)\quad \text{ for } \text{ all } \quad y \in \mathbb {R}. \end{aligned}$$

(28)

Statement (27) was proved earlier in Bening and Korolev [3, Lemma 2.2].

In [9, Proposition 1] the following inequality was proved for $r >0$:

$$\begin{aligned} \sup \limits _{y \ge 0} \left| \mathbb {P} \left( \frac{N_n(r)}{g_n} \le y\right) - G_{r,r}(y) - \frac{h_{2;r}(y)\,{\mathbb {I}}_{\{r > 1\}}(r)}{n} \right| \le \frac{C_2(r)}{n^{\min \{r , 2\}}}, \end{aligned}$$

(29)

where    $h_{2;r}(y) = \frac{\displaystyle 1}{\displaystyle 2\,r}\, g_{r,r}(y) \left( (y-1)(2 - r) + 2 Q_1\big (g_n \,y\big )\right)$    for    $r > 1$,

$$\begin{aligned} Q_1(y) = 1/2 - (y - [y]) \quad \text {and}\,\,\,[y]\,\,\text { is the integer part of a value }y. \end{aligned}$$

(30)

Both Bening, Galieva and Korolev [2] and Gavrilenko, Zubov and Korolev [13] showed the rate of convergence in (29) for $r \le 1$. In Christoph, Monakhov and Ulyanov [7, Theorem 1] the Chebyshev-Edgeworth expansion (29) for $r>1$ is proved.

Remark 4

The random dimension $N_n(r)$ satisfies Assumption 2 of the Transfer Propositions 2 with $g_n=\mathbb {E}N_n(r)$, $H(y)=G_{r,r}(y)$, $h_2(y) =h_{2;r}(y)$ and $b=2$.

In (21), negative moment $\mathbb {E}(N_n(r))^{-a}$ is required where $m^{-a}$ is rate of convergence of Chebyshev-Edgeworth expansion for $T_m$ in (17). Negative moments $\mathbb {E}(N_n(r))^{-a}$ fulfill the estimate:

$$\begin{aligned} \mathbb {E}\big (N_n(r)\big )^{- a} \le C(a, r) \left\{ \begin{array}{ll} n^{- \min \{r, \, a\}}, r \not = a\\ \ln (n) \, n^{- a}, r = a \end{array} \right. \quad \text{ for } \text{ all } \quad r>0 \quad \text{ and } \quad a > 0. \end{aligned}$$

(31)

For $r = a$ the factor $\ln \,n$ cannot be removed. In Christoph, Ulyanov and Bening [10, Corollary 4.2] leading terms for the negative moments of $\mathbb {E}\big (N_n(r)\big )^{- p}$ were derived for any $p>0$ that lead to (31).

The expansions of the length of the vector $Z_m$ in (14) as well as of the sample correlation coefficient $R_n$ in (15) and the angle $\theta _m$ in (16) have as limit distribution the standard normal distribution $\varPhi (x)$. Therefore, with $g_n= \mathbb {E}N_n(r)$ and $\gamma \in \{1/2,\, 0,\, -1/2\}$, limit distributions for

$$\mathbb {P}\Big (g_n^{\gamma } (N_n(r))^{1/2 - \gamma } Z_{N_n(r)} \le x\Big ) \quad \text{ are } \quad G_\gamma (x,r) =\int \nolimits _0^\infty \varPhi (x\,y^\gamma ) \mathrm{d}G_{r,r}(y).$$

These scale mixtures distributions $G_\gamma (x,r)$ are calculated in Christoph and Ulyanov [9, Theorems 3–5]. We apply Proposition 2 to the statistics

$$T_{N_n(r)} = N_n(r)^{1/2 - \gamma }\, Z_{N_n(r)} \quad \text{ with } \text{ the } \text{ normalizing } \text{ factor } \quad g_n^{\gamma }=\mathbb {E}(N_n(r))^\gamma .$$

The limit distributions are:

• for $\gamma = 1/2$ and $r>0$ the Student’s t-distribution $S_{2\,r}(x)$ with density

  $$\begin{aligned} s_{2\,r}(x) = \frac{\displaystyle \varGamma (r+1/2)}{\displaystyle \sqrt{2\,r \pi } \, \varGamma (r)}\,\, \Big (1 + \frac{\displaystyle x^2}{\displaystyle 2\,r}\Big )^{- (r+1/2)}, \quad x \in \mathbb {R}, \end{aligned}$$

  (32)

• for $\gamma =0$ the normal law $\varPhi (x)$,

• for $\gamma = - 1/2$ and $r=2$ the generalized Laplace distributions $L_2(x)$ with density $l_2(x)$:

  $$L_2(x)= \frac{1}{2} + \frac{1}{2}\, \mathrm{sign}(x)\,(1 - (1 + |x|)\, e^{- 2\,|x|} ) \quad \text{ and } \quad l_2(x) = \left( \frac{\displaystyle 1}{\displaystyle 2} + |x|\right) \, e^{- 2\,|x|}.$$

  For arbitrary $r>0$ Macdonald functions $K_{r-1/2}(x)$ occur in the density $l_r(x)$, which can be calculated in closed form for integer values of r.

The standard Laplace density with variance 1  is  $l_1(x) = \frac{\displaystyle 1}{\displaystyle \sqrt{2}} \, {\mathrm{e}}^{- \sqrt{2}\,|x|}$.

Theorem 1

Let $Z_m$ and $N_n(r)$ with $r>0$ be defined by (10) and (1), respectively. Suppose that (14) is satisfied for $Z_m$ and (29) for $N_n(r)$. Then the following statements hold for all $n \in \mathbb {N}_+$:

• (i) Student’s t approximation using scaling factor $\sqrt{\mathbb {E}N_n(r)}$ by $Z_{N_n(r)}$

  $$\begin{aligned} \sup \nolimits _{x} \left| \mathbb {P}\left( \sqrt{g_n}\,Z_{N_n(r)}\le x\right) - S_{2r; n}(x) \right| \le C_r \left\{ \begin{array}{ll} n^{- \min \{r, 3/2\}}, &{} r \not = 3/2, \\ \ln (n) \, n^{- 3/2}, &{} r = 3/2, \end{array} \right. \end{aligned}$$

  (33)

  where

  $$\begin{aligned} S_{2r; n}(x)= & {} S_{2 r}(x) + \, s_{2 r}(x) \left( \frac{\sqrt{2}\,((2r - 5) x^2 - 8 r)}{12\,(2 r - 1) \sqrt{g_n}}\right. \mathbb {I}_{\{r> 1/2\}}(r)\nonumber \\&\,\, \left. + \,\frac{96 r^2 x + (-64 r^2 + 128 r) x^3 + (4 r^2 - 32 r + 39) x^5}{(x^2 + 2 r) (2 r - 1)\,g_n} \mathbb {I}_{\{r > 1\}}(r)\right) , \end{aligned}$$

  (34)

• (ii) Normal approximation with random scaling factor $N_n(r)$ by $Z_{N_n(r)}$

  $$\begin{aligned} \sup \nolimits _{x} \left| \mathbb {P}(\sqrt{N_n(r)}\,Z_{N_n(r)} \le x) - \varPhi _{n\,,\,2}(x)\right| \le C_r \left\{ \begin{array}{ll} n^{- \min \{r, 3/2\}}, &{} r \not = 3/2, \\ \ln (n) \, n^{- 3/2}, &{} r = 3/2, \end{array}\right. \end{aligned}$$

  (35)

  where

  $$\begin{aligned} \varPhi _{n\,,\,2}(x)= & {} \varPhi (x) + \frac{\sqrt{2}\, r \, \varGamma (r - 1/2)}{12\,\varGamma (r)\,\sqrt{g_n}} \,(x^2 -4) \varphi (x)\,\mathbb {I}_{\{r>1/2\}}(r) \nonumber \\& \,+ \,\frac{x^5 - 16 \,x^3 + 24\,x}{144\,g_n}\,\left( \frac{r}{r-1} \,\mathbb {I}_{\{r>1\}}(r)\,+\,\ln \,n \, \mathbb {I}_{\{r=1\}}(r)\right) . \end{aligned}$$

  (36)

• (iii) Generalized Laplace approximation if $r=2$ with mixed scaling factor $g_n^{- 1/2}\,N_n(2)$ by $Z_{N_n(2)}$

  $$\begin{aligned} \sup \nolimits _{x} \left| \mathbb {P}\left( g_n^{- 1/2}\,N_n(2)\,Z_{N_n(2)}\,\le \,x\right) - L_{n;2}(x) \right| \le C_2\, n^{- 3/2} \end{aligned}$$

  (37)

  where

  $$\begin{aligned} L_{n;2}(x)= & {} L_2(x) - \frac{1}{3\,\sqrt{g_n}} \,\left( \frac{1}{\sqrt{2}} + \sqrt{2} |x| - \,x^2\right) \,{\mathrm{e}}^{- 2 |x|}\nonumber \\&\,+ \,\frac{1}{33\,g_n}\,( 12\,\sqrt{2}\,x - 15 |x|\,x +2\,x^3 )\, {\mathrm{e}}^{- 2 |x|}\,. \end{aligned}$$

  (38)

5 The Random Dimension $N_n(s)$ is Discrete Pareto-Like Distributed

The Pareto-like distributed dimension $N_n(s)$ has probability mass function (2) and $\mathbb {E}(N_n(s))=\infty$. Hence $g_n = n$ is chosen as normalizing sequence for $N_n(s)$.

Bening and Korolev [4, Sect. 4.3] showed that for integer $s \ge 1$

$$ \lim \nolimits _{n \rightarrow \infty } \sup \nolimits _{y>0}|\mathbb {P}(N_n(s)\le n\,y) - H_s(y)| =0. $$

where $H_s(y) = {\mathrm{e}}^{-s/y} {\mathbb {I}}_{(0\,,\,\infty )}(y)$ is the continuous distribution function of the inverse exponential $W(s)=1/V(s)$ with exponentially distributed V(s) having rate parameter $s>0$. As $\mathbb {P}(N_n(s) \le y)$, so $H_s(y)$ is heavy tailed with shape parameter 1 and $\mathbb {E}W(s) = \infty$.

Lyamin [18] proved a bound $|\mathbb {P}(N_n(s)\le n\,y) - H_s(y)| \le C/n$ and $C< 0.37$ for integer $s \ge 1$.

In [9, Proposition 2] the following results are presented for $s >0$:

$$\begin{aligned} \sup _{y>0}\left| \mathbb {P}\left( \frac{N_n(s)}{n} \le y\right) - H_s(y) - \frac{h_{2;s}(y)}{n}\right| \le \frac{C_3(s)}{n^2}, \quad \text{ for } \text{ all } \quad n \in \mathbb {N}_+, \end{aligned}$$

(39)

with $H_s(y) = {\mathrm{e}}^{-s/y}$    and    $h_{2;s}(y) = s\, {\mathrm{e}}^{-s/y}\,\big (s-1 + 2 Q_1(n\,y)\big )/\big (2\,y^2\big )$  for   $y>0$, where $Q_1(y)$ is defined in (30). Moreover

$$\begin{aligned} \mathbb {E}\big (N_n(s)\big )^{- p} \le C(p) \, n^{- \min \{p, 2\}}, \end{aligned}$$

(40)

where for $0< p \le 2$ the order of the bound is optimal.

The Chebyshev-Edgeworth expansion (39) is proved in Christoph, Monakhov and Ulyanov [7, Theorem 4]. The leading terms for the negative moments $\mathbb {E}\big (N_n(s)\big )^{- p}$ were derived in Christoph, Ulyanov and Bening [10, Corollary 5.2] that lead to (40).

Remark 5

The random dimension $N_n(s)$ satisfies Assumption 2 of the Transfer Propositions 2 with $H_s(y) = {\mathrm{e}}^{-s/y}$,  $h_2(y) =h_{2;s}(y)$,  $g_n=n$ and $b=2$.

With $g_n= n$ and $\gamma \in \{1/2,\, 0,\, -1/2\}$, the limit distributions for

$$\mathbb {P}\left( n^{\gamma } N_n(s)^{1/2 - \gamma } Z_{N_n(s)} \le x\right) \quad \text{ are } \text{ now } \quad G_\gamma (x,s) =\int \nolimits _0^\infty \varPhi (x\,y^\gamma ) {\mathrm{d}} H_s(y).$$

These scale mixtures distributions $G_\gamma (x,s)$ are calculated in Christoph and Ulyanov [9, Theorems 6–8]. We apply Proposition 2 to statistics

$$T_{N_n(s)} = N_n(s)^{1/2 - \gamma }\, Z_{N_n(s)}\quad \text{ with } \text{ the } \text{ normalizing } \text{ factor } \,\,\, n^{\gamma }.$$

The limit distributions are:

• for $\gamma = 1/2$   Laplace distributions $L_{1/\sqrt{s}}(x)$ with density

  $$l_{1/\sqrt{s}}(x)=\sqrt{s/2}\,{\mathrm{e}}^{-\sqrt{2\,s}|x|},$$

• for $\gamma =0$ the standard normal law $\varPhi (x)$ and

• for $\gamma = - 1/2$ the scaled Student’s t-distribution $S_2^*(x; \sqrt{s})$ with density

  $$s^*_{2}(x; \sqrt{s})= \frac{\displaystyle 1}{\displaystyle 2\,\sqrt{2\,s}}\Big (1 + \frac{\displaystyle x^2}{\displaystyle 2\,s}\Big )^{- 3/2}.$$

Theorem 2

Let $Z_m$ and $N_n(s)$ with $s>0$ be defined by (10) and (2), respectively. Suppose that (14) is satisfied for $Z_m$ and (39) for $N_n(s)$. Then the following statements hold for all $n \in \mathbb {N}_+$:

• (i) Laplace approximation with non-random scaling factor $n^\gamma$ by $Z_{N_n(s)}$:

  $$\begin{aligned} \sup \nolimits _{x} \left| \mathbb {P}\left( \sqrt{n}\,Z_{N_n(s)}\le x\right) - L_{1/\sqrt{s}; n}(x) \right| \le C_s\, n^{- 3/2} \end{aligned}$$

  (41)

  where

  $$\begin{aligned} L_{1/\sqrt{s}; n}(x)= & {} L_{1/\sqrt{s}}(x) + \, l_{1/\sqrt{s}}(x)\left( \frac{\sqrt{2}}{12\,s\,\sqrt{n}}\,(s x^2 - 2 \,(1 + \sqrt{2\,s\, }\,|x|)\right. \nonumber \\&\left. \,+\,\frac{s}{72 \, n}\left( \frac{x^3\,|x|}{\sqrt{2\,s}} - \frac{8\, x^2}{s} + \frac{6\,x}{s^2} \,(1 + \sqrt{2\,s\,}\,|x|)\,\right) \right) \end{aligned}$$

  (42)

  ,

• (ii) Normal approximation with random scaling factor $\sqrt{N_n(s)}$ by $Z_{N_n(r)}$:

  $$\begin{aligned} \sup \nolimits _{x} \left| \mathbb {P}\left( \sqrt{N_n(s)}\,Z_{N_n(s)}\,\le \,x\right) - \varPhi _{n, 2}(x)\right| \le C_s \, n^{- 3/2}, \end{aligned}$$

  (43)

  where

  $$\begin{aligned} \varPhi _{n, 2}(x) = \varPhi (x) + \varphi (x) \left( \frac{\sqrt{2\,\pi } (x^2 - 4)}{24 \, \sqrt{n}} + \frac{x^5 - 16 x^3 +24 x}{144\,s\, n} \right) \end{aligned}$$

  (44)

• (iii) Scaled Student’s t-distribution with mixed scaling factor by $Z_{N_n(s)}$

  $$\begin{aligned} \sup \nolimits _{x} \left| \mathbb {P}\left( n^{- 1/2}\,N_n(s)\,Z_{N_n(s)}\,\le \,x\right) - S^*_{n;2}(x) \right| \le C_s \, n^{- 3/2}, \end{aligned}$$

  (45)

  where

  $$\begin{aligned} S^*_{n;2}(x;\sqrt{s})= & {} S^*_2(x;\sqrt{s}) + s_2^*(x; \sqrt{s})\left( -\frac{\sqrt{2}\,(x^2 + 8\,s)}{12 (2\,s + x^2) \,\sqrt{n}}\,\right. \nonumber \\&\left. + \,\frac{1}{144 \,n}\left( \frac{105 x^5}{(2\,s + x^2)^3} + \frac{240 x^3}{(2\,s + x^2)^2} + \frac{72 x}{2\,s + x^2}\right) \right) . \end{aligned}$$

  (46)

6 Proofs of Main Results

Proof

The proofs of Theorems 1 and 2 are based on Proposition 2. The structure of the functions $f_1$, $f_2$ and $h_2$in Assumptions A and B is similar to the structure of the corresponding functions in Conditions 1 and 2 in [9]. Therefore, the estimates of the term $D_n$ and of the integrals $I_1(x,n)$ and $I_2(x,n)$ in (23), (25) and (24) as well as the validity of (19) and (20) in Proposition 2 when H(y) is $G_{r,r}(y)$ or $H_s(y)$ can be shown analogously to the proofs for Lemmas 1, 2 or 4 in [9]. In Remark 3 above it was pointed out that the integrals in (25) and (25) can degrade the convergence rate. Let $r<1$. With $|f_2(x\,y^\gamma | \le c^*$ we get

$$\begin{aligned} \int \nolimits _{1/g_n}^\infty \frac{|f_2(x\,y^\gamma )|}{g_n\,y}\mathrm{d}G_{r,r}(y) \le \frac{c^*r^r}{\varGamma (r)\,g_n} \int \nolimits _{1/g_n}^\infty y^{r-2} \mathrm{d}y \le \frac{c^*r^r}{(1-r)\varGamma (r)} \,g_n^{- r}. \end{aligned}$$

(47)

The additional term $f_1(xy^{\gamma })\,(g_n y)^{- 1/2}$ in (17) in Assumption A is to be estimated with condition (19ii).

Moreover, the bounds for $\mathbb {E}(N_n)^{-3/2}$ follow from (31) and (40), since $a=3/2$ in Assumption A, considering the approximation (14).

The integrals in (22) in Proposition 2 are still to be calculated. Similar integrals are calculated in great detail in the proofs of Theorems 3–8 in [9]. To obtain (34), we compute the integrals with Formula 2.3.3.1 in Prudnikov et al. [20]

$$\begin{aligned} M_{\alpha }(x) = \frac{r^r}{\varGamma (r)\,\sqrt{2 \pi }}\,\int \limits ^{\infty }_{0}y^{\alpha -1} {\mathrm{e}}^{- (r+x^2/2)y} dy = \frac{\varGamma (\alpha ) \, r^{r-\alpha } }{\varGamma (r)\,\sqrt{2\, \pi }}\, \big (1+x^2/(2r)\big )^{- \alpha }\, \end{aligned}$$

(48)

for   $\alpha = r-1/2, \,\, r~+~1/2,\,\, r~+~3/2$ and $p=r+x^2/2$.

Lemma 2 in [9] and $\int \nolimits _0^\infty y^{- 1} d G_{r,r}(y) = r/(r-1)$ for $r>1$ lead to (36).

To show (38) we use Formula 2.3.16.2 in [20] with $n=0, 1$ and Formula 2.3.16.3 in [20] with $n=1, 2$ and $p=2$ and $q=x^2/2$.

To obtain (42), we calculate the integrals again with Formula 2.3.16.3 in [20], with $p=x^2/2>0$, $q=s>0$, $n =0, 1, 2$.

Lemma 4 in [9] and $\int \nolimits _0^\infty y^{- a-1} {\mathrm{e}}^{-s/y} d y = s^{-a} \varGamma (a)$ for $a=3/2,\,2$ lead to (44).

Finally, in $\int \nolimits _0^\infty f_k(x/\,y^\gamma ) y^{-2-k/2} {\mathrm{e}}^{-s/y}\mathrm{d}y$ we use the substitution $s/y = u$ to obtain, with (48), the terms in (46).    $\square$