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Mathematics
G. N. SYTAYA
ON THE LIMITING DISTRIBUTION OF A CERTAIN CLASS OF FUNCTIONALS OF A SEQUENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES
(Presented by Academician A. N. Kolmogorov, 7 IV 1964)
Let \(\xi_1, \xi_2, \ldots, \xi_n, \ldots\) be a sequence of independent identically distributed random variables with \(M\xi_n = 0,\ D\xi_n = 1\). Denote
\[ S_{nk}=\frac{1}{\sqrt n}\sum_{i=1}^{k}\xi_i;\qquad S_{n0}=0. \]
Consider a sequence of nonnegative measurable functions \(\Phi_n(x,y)\), defined for \(x\in(-\infty,\infty)\), \(y\in(-\infty,\infty)\). We shall seek conditions sufficient for the existence of a limiting distribution of the quantities
\[ \eta_n=\sum_{j=0}^{n-1}\sum_{k=0}^{n-1}\Phi_n(S_{nj},S_{nk}). \]
If \(\Phi(x,y)\) is a continuous function, then from M. Donsker’s theorem \((^1)\) it follows that the limiting distribution of the quantities
\[ \eta_n=\frac{1}{n^2}\sum_{j=0}^{n-1}\sum_{k=0}^{n-1}\Phi(S_{nj},S_{nk}) \]
exists and coincides with the distribution of the functional
\[ \int_0^1\int_0^1 \Phi(w(t),w(s))\,dt\,ds \]
(\(w(t)\) is the Brownian motion process).
For arbitrary nonnegative measurable functions \(\Phi_n(x)\) such that
\[ \lim_{n\to\infty}\sup \Phi_n(x)=0, \]
the problem was considered by A. V. Skorokhod in the paper \((^2)\), in which it is established that, for the existence of a limiting distribution of the quantities
\[ \eta_n=\sum_{j=0}^{n-1}\Phi_n(S_{nj}), \]
it is sufficient that: 1) there exist a square-integrable density of the random variables \(\xi_j\), and 2) the sequence of functions
\[ u_n(x)=2n\int_0^x \Phi_n(z)\,dz \]
converge almost everywhere to some limit \(u(x)\). In this case the limiting distribution of the quantities \(\eta_n\) coincides with the distribution of the random variable
\[ \int_0^{w(1)} u(x)\,dx-\int_0^1 u(w(s))\,dw(s). \]
This result was generalized to the case of sign-changing functions by N. P. Slobodenyuk \((^3)\).
On the basis of the results of article \((^2)\), one can prove the validity of the following assertion:
Theorem. Suppose that the following hold:
- The \(\xi_j\) have a square-integrable density \(p(x)\),
\[ \int |x|^3 p(x)\,dx < \infty . \]
\[ \lim_{n\to\infty}\sup_{x,y} n^{3/2}\Phi_n(x,y)=0. \]
- For each \(N\), the quantities \(n u_n(x,y)\), \(n u_n^*(x,y)\), where
\[ u_n(x,y)=2n\int_0^x \Phi_n(\alpha,y)\,d\alpha;\qquad u_n^*(x,y)=2n\int_0^y \Phi_n(x,\beta)\,d\beta, \]
are uniformly bounded in the aggregate on the set \(|x|\le N;\ |y|\le N\).
- The sequence
\[ q_n(x,y)=4n^2\int_0^x\int_0^y \Phi_n(\alpha,\beta)\,d\alpha\,d\beta \]
converges almost everywhere to the function \(q(x,y)\).
Then the limiting distribution of \(\eta_n\) exists and coincides with the distribution of the random variable
\[ \int_0^{w(1)}\int_0^{w(1)} q(x,y)\,dx\,dy -\int_0^1\int_0^{w(1)} q(x,w(s))\,dx\,dw(s) - \]
\[ -\int_0^{w(1)}\int_0^1 q(w(t),y)\,dw(t)\,dy +\int_0^1\int_0^1 q(w(t),w(s))\,dw(t)\,dw(s), \tag{1} \]
where
\[ \int_0^1\int_0^{w(1)} q(x,w(s))\,dx\,dw(s) = p\lim_{n\to\infty}\sum_{k=0}^{n-1}\int_0^{w(1)} q(x,w(t_k))\,dx\,\Delta w(t_k), \]
\[ \int_0^1\int_0^1 q(w(t),w(s))\,dw(t)\,dw(s) = p\lim_{n\to\infty}\sum_{j=0}^{n-1}\sum_{k=0}^{n-1} q(w(t_j),w(t_k))\,\Delta w(t_j)\Delta w(t_k). \]
The proof of the theorem is based on the following lemmas.
Lemma 1. Let \(\Phi_n(x,y)=0\) for \(|x|>c\) or \(|y|>c\). Then, if conditions 1 and 2 of the theorem are satisfied and the functions
\[ n\bar u_n(x,y)=2n^2\int_{-\infty}^{x}\Phi_n(\alpha,y)\,d\alpha \]
are bounded in the aggregate, the quantities \(\eta_n\) have a limiting distribution if and only if there exists a limiting distribution of the quantities
\[ \sum_{k=0}^{n-1}\int_0^{S_{nn}}\bar u_n(x,S_{nk})\,dx - \sum_{j=0}^{n-1}\sum_{k=0}^{n-1} \bar u_n(S_{nj}S_{nk})\frac{\xi_{j+1}}{\sqrt n}. \tag{2} \]
and the two distributions coincide.
Lemma 2. If the conditions of Lemma 1 are satisfied and the quantities
\[ n\bar u_n^*(x,y)=2n^2\int_{-\infty}^{y}\Phi_n(x,\beta)\,d\beta \]
are bounded in the aggregate, then the limiting distribution of (2), and consequently also of \(\eta_n\), exist simultaneously, and if (2) exists, then it coincides
with limiting distribution
\[ \int_{0}^{S_{nn}}\int_{0}^{S_{nn}} \overline q_n(x,y)\,dx\,dy -\sum_{k=0}^{n-1}\int_{0}^{S_{nn}} \overline q_n(x,S_{nk})\,dx\,\frac{\xi_{k+1}}{\sqrt n} -\sum_{j=0}^{n-1}\int_{0}^{S_{nn}} \overline q_n(S_{nj},y)\,dy\,\frac{\xi_{j+1}}{\sqrt n} +\sum_{j=0}^{n-1}\sum_{k=0}^{n-1}\overline q_n(S_{nj},S_{nk})\frac{\xi_{j+1}}{\sqrt n}\frac{\xi_{k+1}}{\sqrt n}, \tag{3} \]
where
\[ \overline q_n(x,y)=4n^2\int_{-\infty}^{x}\int_{-\infty}^{y}\Phi_n(\alpha,\beta)\,d\alpha\,d\beta . \]
Lemma 3. If the conditions of Lemmas 1 and 2 are satisfied and the sequence \(\overline q_n(x,y)\) converges almost everywhere to a nondecreasing function \(\overline q(x,y)\), then the limiting distribution of the quantity (3) exists and coincides with the distribution of the random variable
\[ \int_{0}^{w(1)}\int_{0}^{w(1)}\overline q(x,y)\,dx\,dy -\int_{0}^{1}\int_{0}^{w(1)}\overline q(x,w(s))\,dx\,dw(s) \]
\[ -\int_{0}^{w(1)}\int_{0}^{1}\overline q(w(t),y)\,dw(t)\,dy +\int_{0}^{1}\int_{0}^{1}\overline q(w(t),w(s))\,dw(t)\,dw(s). \]
Example. Suppose that, for \(\xi_1,\xi_2,\ldots,\xi_n,\ldots\), condition 1 of the theorem is satisfied. Consider the function
\[ \Phi_n(x,y) \begin{cases} \dfrac{1}{n^2\varepsilon_n}, & \text{when } |x-y|<\varepsilon_n,\\[6pt] 0, & \text{when } |x-y|\ge \varepsilon_n . \end{cases} \]
It is not difficult to verify that if \(\varepsilon_n\) is chosen equal to \(n^{-1/2+\delta}\) \((0<\delta<1/2)\), then conditions 2–4 of the theorem are satisfied, and the sequence \(q_n(x,y)\) converges to the value
\[ q(x,y)= \begin{cases} 0, & \text{when } x\cdot y \le 0,\\ 8\min(|x|,|y|), & \text{when } x\cdot y>0. \end{cases} \]
Consequently, there exists a limiting distribution of the functional \(\eta_n\), coinciding with the distribution of the random variable (1). In the present case formula (1), with the aid of K. Ito’s theorem\({}^{4}\), can be transformed into the form
\[ 4\left\{\int_{0}^{1}|w(t)-w(1)|\,dt-\int_{0}^{1}|w(t)|\,dt +w(1)\int_{0}^{1}\operatorname{sgn}w(t)\,dt\right\} \]
\[ -\frac{8}{3}|w(1)|^3+\int_{0}^{1}\int_{0}^{1}q(w(t),w(s))\,dw(t)\,dw(s). \]
In conclusion I express my sincere gratitude to A. V. Skorokhod, who posed the problem and guided its solution.
Institute of Mathematics
Academy of Sciences of the USSR
Received
7 IV 1964
REFERENCES
\({}^{1}\) M. Donsker, Mem. Am. Math. Soc., 2, No. 6 (1951).
\({}^{2}\) A. V. Skorokhod, Ukr. Math. Zhurn., 13, No. 4, 67 (1961).
\({}^{3}\) N. P. Slobodenyuk, Dop. AN URSR, No. 6 (1963).
\({}^{4}\) K. Ito, Sborn. per. Matematika, 1, 1 (1957).