Multivariate order statistics based on dependent and nonidentically distributed random variables

In this paper some identities and inequalities which involve the joint distribution of order statistics in a set of dependent and nonidentically distributed random variables are derived. These identities and inequalities provide a unified way to handle the joint distribution of order statistics in a set of univariate or bivariate observations.     

Computing the moments of order statistics from nonidentically distributed phase-type random variables

In this paper, we derive a recurrence relation for the single moments of order statistics (o.s.) arising from n independent nonidentically distributed phase-type (PH) random variables (r.v.’s). This recurrence relation will enable one to compute all single moments of all o.s. in a simple recursive manner.     

Rate of convergence to the normal law of sums of induced order statistics

One obtains a certain estimate of order 1/n of the rate of convergence to the normal law for induced order statistics. The result includes the multidimensional case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 108, pp. 45–56, 1981.     

Rate of convergence of the joint distribution of order statistics to the normal law

Bounds are derived on the rate of convergence of the joint distribution of order statistics to the corresponding multivariate normal distribution.     

Record and interrecord times for sequences of nonidentically distributed random variables

Assume that the independent random variables X₁,X₂,... have the distribution functions , ..., respectively, where F is an arbitrary continuous distribution function, while _i are positive constants. In this situation, one obtains some theorems for the record moments and interrecord times.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 109–118, 1985.     

A limit theorem for sums of independent,nonidentically distributed random variables

A generalization and refinement of Chen's theorem related to a strong law of large numbers for sums of independent, nonidentically distributed random variables are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta in im. V. A. Steklova AN SSSR, Vol. 85, pp. 188–192, 1979.     

Recurrence relations for order statistics from n independent and non-identically distributed random variables

Some well-known reeurrence relations for order statistics in the i.i.d. case are generalized to the case when the variables are independent and non-identically distributed. These results could be employed in order to reduce the amount of direct computations involved in evaluating the moments of order statistics from an outlier model.     

The number of records in a sequence of nonidentically distributed random variables

One obtains limit theorems for the number of records and for the times of the attainment of the record values in a sequence of independent random variables.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 373–388, 1986.     

Estimates of the rate of convergence of sums of order statistics to the normal law. II

Some results are obtained on the rate of convergence of trimmed means to the normal Law. The work continues the investigations begun in the paper [3].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 55, pp. 165–174, 1976.     

The geometric convergence of processes,generated by a sequence of nonidentically distributed variables

In the paper one investigates tests for the weak convergence of the distribution of random closed sets, corresponding to a sequence of independent, identically distributed random variables.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 81–85, 1985.     

非同分布NA序列的完全收敛性

讨论了非同分布NA序列部分和与随机足标部分和的完全收敛性，推广了于浩在1989年得到的关于独立随机变量序列的一些结果。     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

Empirical distribution functions and functions of order statistics for mixing random variables

Some fundamental properties of the empirical distribution functions are derived in the case of mixing random variables. These properties are then utilized to study asymptotic normality and strong laws of large numbers for functions of order statistics.