Asymptotic notation for the execution time of programs. Estimates from below, from above, asymptotically exact. The sum rule and the product rule. Asymptotic behavior of functions. comparison of infinitesimal functions Asymptotic selection criteria

The object of research and the relevance of the topic. In the theory of statistical analysis of discrete sequences, a special place is occupied by goodness-of-fit tests for testing the possibly complex null hypothesis, which is that for a random sequence such that

Xi e hi,i = 1, ,n, where hi = (0,1,. ,M), for any i = 1,., n, and for any k £ 1m the probability of the event

Xi = k) does not depend on r. This means that the sequence is in some sense stationary.

In a number of applied problems, the sequence (Xr-)™=1 is considered to be the sequence of colors of balls when choosing without returning to exhaustion from an urn containing u - 1 > 0 balls of color k, k € 1m- 1, .,pm - 1). Let the urn contain n - 1 balls, m k=0

Denote by r(k) (fc) Jk) rw - Г! , . . . , the sequence of numbers of balls of color A; in the sample. Consider the sequence where k)

Kk-p-GPk1.

The sequence h^ is defined using the distances between places of adjacent balls of color k in such a way that

Pk Kf \u003d p. 1> \u003d 1

The set of sequences h(fc) for all k £ 1m uniquely determines the sequence The sequences hk for different k are mutually dependent. In particular, any of them is uniquely determined by all the others. If the cardinality of the set 1m is equal to 2, then the sequence of colors of the balls is uniquely determined by the sequence of distances between the places of adjacent balls of the same fixed color. Let an urn containing n - 1 balls of two different colors contain N - 1 balls of color 0. One can establish a one-to-one correspondence between the set ffl(N - l,n - N) and the set 9\n,N of vectors h(n, N ) = (hi,., hjf) with positive integer components such that K = P. (0.1)

The set 9p)dz corresponds to the set of all different partitions of the whole positive number n into N ordered terms.

Having given some probability distribution on the set of vectors £Hn,dz, we obtain the corresponding probability distribution on the set Wl(N - 1,n - N). The set is a subset of the set of vectors with non-negative integer components satisfying (0.1). As probability distributions on a set of vectors in the dissertation work, distributions of the form

P(t,N) = (n,.,rN)) = P(tn = ru,v = l,.,N\jr^ = n), (0.2) where. , £dz - independent non-negative integer random variables.

Distributions of the form (0.2) in /24/ are called generalized schemes for placing n particles in N cells. In particular, if the random variables £b. , £n in (0.2) are distributed according to the Poisson laws with the parameters Ai,., λ, respectively, then the vector h(n,N) has a polynomial distribution with the probabilities of outcomes

Ri = . , A" ,V = \,.,N.

L\ + . . . +AN

If the random variables £b >&v in (0-2) are equally distributed according to the geometric law where p is any in the interval 0< р < 1, то, как отмечено в /25/,/26/, получающаяся обобщенная схема размещения соответствует равномерному распределению на множестве В силу взаимнооднозначного соответствия между множеством dft(N - 1 ,п - N) и множеством tRn,N получаем равномерное распределение на множестве выборов без возвращения. При этом, вектору расстояний между местами шаров одного цвета взаимно однозначно соответствует вектор частот в обобщенной схеме размещения, и, соответственно, числу расстояний длины г - число ячеек, содержащих ровно г частиц. Для проверки по единственной последовательности гипотезы о том, что она получена как результат выбора без возвращения, и каждая такая выборка имеет одну и ту же вероятность можно проверить гипотезу о том, что вектор расстояний между местами шаров цвета 0 распределен как вектор частот в соответствующей обобщенной схеме размещения п частиц по N ячейкам.

As noted in /14/,/38/, a special place in testing hypotheses about the distribution of frequency vectors h(n, N) = (hi,., /rz) in generalized schemes for placing n particles in N cells is occupied by criteria built on based on statistics of the form 1 m(N -l,n-N)\ N

LN(h(n,N))=Zfv(hv)

Fn \u003d F (-T7, flQ Hi II-

0.4) where fu, v = 1,2,. and φ are some real-valued functions, N

Mr = E = r), r = 0.1,. 1/=1

The values ​​in /27/ were called the number of cells containing exactly r particles.

Statistics of the form (0.3) in /30/ are called separable (additively separable) statistics. If the functions /n in (0.3) do not depend on u, then such statistics were called in /31/ symmetric separable statistics.

For any r, the statistic /xr is a symmetric separable statistic. From equality

E DM = E DFg (0.5) it follows that the class of symmetric separable statistics in hv coincides with the class linear functions from fir. Moreover, the class of functions of the form (0.4) is wider than the class of symmetric separable statistics.

But = (#o(n, N)) is a sequence of simple null hypotheses that the distribution of the vector h(n,N) is (0.2), where random variables,. in (0.2) are identically distributed and k) = pk,k = 0,1,2,., the parameters n and N vary in the central region.

Let us consider some Р £ (0,1) and a sequence of, generally speaking, complex alternatives

H \u003d (H (n, N)) such that there is - maximum number, for which, for any simple hypothesis H\ ∈ H(n, N), the inequality

РШ > an,N(P)) > Р

We will reject the hypothesis Hq(ti,N) if fm > awm((3). If there exists a limit

Wn ~1nP(0lg > an,N(P))=u(p,Н), where the probability for each N is calculated under the hypothesis Нц(п, N), then the value of ^(/З, Н) is named in /38/ criterion index φ at the point (j3, H). The last limit may, generally speaking, not exist. Therefore, in the dissertation work, in addition to the criterion index, the value

Ish (~1pR(fm > al(/?)))

JV->oo N-oo mean respectively the lower and upper limits of the sequence (odr) when N -> oo,

If a criterion index exists, then the subscript of the criterion matches it. The subscript of the criterion always exists. The greater the value of the criterion index (lower index of the criterion), the better the statistical criterion in the considered sense. In /38/ the problem of constructing goodness-of-fit criteria for generalized layouts with highest value index of the criterion in the class of criteria that reject the hypothesis Ho(n,N) at /MO Ml Mtch HF iV" iV""""" ~yv" " ^ " where m > 0 is some fixed number, the sequence of constants from a given value of the power of the criterion for a sequence of alternatives, ft is a real function of m + 1 arguments.

The criteria indices are determined by the probabilities of large deviations. As was shown in /38/, the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of separable statistics under the Cramer condition for the random variable f(t) is determined by the corresponding Kullback-Leibler-Sanov information distance (the random variable rj satisfies the condition Cramer, if for some λ > 0 the moment generating function Metr] is finite in the interval \t\< Н /28/).

The question of the probabilities of large deviations of statistics from an unlimited number of fir, as well as arbitrary separable statistics that do not satisfy the Cramer condition, remained open. This did not make it possible to finally solve the problem of constructing criteria for testing hypotheses in generalized allocation schemes with the highest rate of convergence to zero of the probability of error of the first kind with approaching alternatives in the class of criteria based on statistics of the form (0.4). The relevance of the dissertation research is determined by the need to complete the solution of this problem.

The purpose of the dissertation work is to construct goodness-of-fit criteria with the highest value of the criterion index (lower index of the criterion) for testing hypotheses in the selection scheme without recurrence in the class of criteria that reject the hypothesis W(n, N) at $.<>,■ ■)><*. (0-7) где ф - функция от счетного количества аргументов, и параметры п, N изменяются в центральной области.

In accordance with the purpose of the study, the following tasks were set:

Investigate the properties of entropy and Kullback - Leibler - Sanov information distance for discrete distributions with a countable number of outcomes;

Investigate the probabilities of large deviations of statistics of the form (0.4);

Investigate the probabilities of large deviations of symmetric separable statistics (0.3) that do not satisfy the Cramer condition;

Find a statistic such that the goodness-of-fit criterion constructed on its basis for testing hypotheses in generalized allocation schemes has the largest index value in the class of criteria of the form (0.7).

Scientific novelty:

Scientific and practical value. In the paper, a number of questions about the behavior of large deviation probabilities in generalized allocation schemes are solved. The results obtained can be used in the educational process in the specialties of mathematical statistics and information theory, in the study of statistical procedures for the analysis of discrete sequences and were used in /3/, /21/ when justifying the security of one class of information systems. Provisions for defense:

Reduction of the problem of checking, by a single sequence of colors of the balls of the hypothesis, from the fact that this sequence was obtained as a result of a choice without replacement until the exhaustion of balls from the urn containing balls of two colors, and each such choice has the same probability, to the construction of goodness-of-fit criteria for testing hypotheses in the corresponding generalized placement scheme;

Continuity of the functions of entropy and Kullback - Leibler - Sanov information distance on an infinite-dimensional simplex with the introduced logarithmic generalized metric;

A theorem on the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of symmetric separable statistics that do not satisfy the Cramer condition in the generalized allocation scheme in the seven exionential case;

A theorem on the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations for statistics of the form (0.4);

Construction of a goodness-of-fit criterion for testing hypotheses in generalized layouts with the largest index value in the class of criteria of the form (0.7).

Approbation of work. The results were reported at the seminars of the Department of Discrete Mathematics of the Mathematical Institute. V. A. Steklov RAS, Department of Information Security ITMiVT them. S. A. Lebedev RAS and at:

Fifth All-Russian Symposium on Applied and Industrial Mathematics. Spring session, Kislovodsk, May 2 - 8, 2004;

Sixth International Petrozavodsk Conference "Probabilistic Methods in Discrete Mathematics" June 10 - 16, 2004;

Second International Conference "Information Systems and Technologies (IST"2004)", Minsk, November 8-10, 2004;

International conference "Modern Problems and new Trends in Probability Theory", Chernivtsi, Ukraine, June 19 - 26, 2005.

The main results of the work were used in the research work "Apologia", carried out by ITMiVT RAS. S. A. Lebedev in the interests of the Federal Service for Technical and Export Control of the Russian Federation, and were included in the report on the implementation of the research stage /21/. Separate results of the dissertation were included in the research report "Development of mathematical problems of cryptography" of the Academy of Cryptography of the Russian Federation for 2004 /22/.

The author expresses his deep gratitude to the scientific adviser, Doctor of Physical and Mathematical Sciences Ronzhin A.F. and scientific consultant, Doctor of Physical and Mathematical Sciences, Senior Researcher Knyazev A.V. Mathematical Sciences I. A. Kruglov for the attention shown to the work and a number of valuable remarks.

Structure and content of the work.

The first chapter investigates the properties of entropy and information distance for distributions on the set of non-negative integers.

In the first paragraph of the first chapter, the notation is introduced and the necessary definitions are given. In particular, the following notation is used: x = (xq, x\, . ) is an infinite-dimensional vector with a countable number of components;

H(x) - -Ex^oXvlnx,-, truncm(x) = (x0,x1,.,xm,0.0,.)] f2* = (x, xi > 0, zy = 0.1,. , Oh"< 1}; Q = {х, х, >0,u = 0,1,., o xv = 1); = (x G 0, ££L0 = 7);

Ml = o Ue>1|5 € o< Ml - 7МГ1 < 00}. Понятно, что множество £1 соответствует семейству вероятностных распределений на множестве неотрицательных целых чисел, П7 - семейству вероятностных распределений на множестве неотрицательных целых чисел с математическим ожиданием 7.

If y 6 E Π, then for e > 0 Oe(y) will denote the set

Oe(y) - (x ^< уие£ для всех v = 0,1,.}.

In the second paragraph of the first chapter, we prove a theorem on the boundedness of the entropy of discrete distributions with bounded mathematical expectation.

Theorem 1. On the boundedness of the entropy of discrete distributions with bounded mathematical expectation.

For any f 6 P7

H(x)

If x € fly corresponds to a geometric distribution with mathematical definition 7, that is, 7

1 + 7 then the equality

H(x) = F(<7).

The assertion of the theorem can be viewed as the result of a formal application of the method of conditional Lagrange multipliers in the case of an infinite number of variables. The theorem that the only distribution on the set (k, k + 1, k + 2,.) with a given mathematical expectation and maximum entropy is a geometric distribution with a given mathematical expectation is given (without proof) in /47/. The author, however, gave a rigorous proof.

In the third paragraph of the first chapter, a definition of a generalized metric is given - a metric that admits infinite values.

For x, y ∈ Q, the function p(x, y) is defined as the minimal e > 0 with the property<хи< уиее для всех и = 0,1,. Если такого е не существует, то полагается, что р(х,у) = оо.

It is proved that the function p(x, y) is a generalized metric on the family of distributions on the set of non-negative integers, as well as on the entire set Cl*. Instead of e in the definition of the metric p(x, y), you can use any other positive number other than 1. The resulting metrics will differ by a multiplicative constant. Denote by J(x, y) the information distance

00 £ J(x, y) = E In-.

Here and below it is assumed that 0 In 0 = 0.0 In jj = 0. The information distance is defined for such x, y that xn = 0 for all and such that yi = 0. If this condition is not satisfied, then we will put J(x,ij) = oo. Let L SP. Then we will denote

J (A Y) = |nf J(x, y).

In the fourth section of the first chapter, a definition is given for the compactness of functions defined on the set Q*. The compactness of a function with a countable number of arguments means that, with any degree of accuracy, the value of the function can be approximated by the values ​​of this function at points where only a finite number of arguments are nonzero. The compactness of the entropy and information distance functions is proved.

1. For any 0< 7 < оо функция Н(х) компактна на

2. If for some 0< 70 < оо

P e then for any 0<7<оо,г>0 the function χ) = J(x, p) is compact on

In the fifth paragraph of the first chapter, the properties of the information distance given on an infinite-dimensional space are considered. Compared to the finite-dimensional case, the situation with the continuity of the information distance function changes qualitatively. It is shown that the information distance function is not continuous on the set in any of the metrics

Pl&V) = E\Xu~Y"\, u=0

E (xv - Yi) 2 v \u003d Q

Pz(x, y) = 8Up\xu-yv\. v

The validity of the following inequalities for the functions of entropy H(x) and information distance J(x,p) is proved:

1. For any x, x" € fi

H(x) - H(x")\< - 1){Н{х) + Н{х")).

2. If for some x, p e Π there exists e > 0 such that x 6 0 £(p), then for any x" £ Q J(x, p) - J(x", p)|< (е"М - 1){Н{х) + Н{х") + ееН(р)).

From these inequalities, taking into account Theorem 1, it follows that the functions of entropy and information distance are uniformly continuous on the corresponding subsets of Q in the metric p(x,y)t, namely,

1. For any 7 such that 0< 7 < оо, функция Н(х) равномерно непрерывна на Г2 в метрике р(ж,у);

2. If for some 70, 0< 70 < оо

TO for any 0<7<оои£>0 function

A p(x) = J(x, p) is uniformly continuous on the set Π Oe(p) in the metric p(x, y).

The definition of non-extremality of a function is given. The non-extremality condition means that the function does not have local extrema, or the function takes the same values ​​in local minima (local maxima). The non-extremality condition weakens the requirement that there are no local extrema. For example, the function sin x on the set of real numbers has local extrema, but satisfies the non-extremality condition.

Let for some 7 > 0, the area A is given by the condition

A = (x € VLv4>(x) > a), (0.9) where φ(x) is a real-valued function, a is some real constant, inf φ(x)< а < inf ф(х).

The question was studied under what conditions on the function φ when changing parameters n,N in the central region, ^ -; 7, for all their sufficiently large values ​​there exist non-negative integers ko, k\,., kn such that k0 + ki + . + kn = N, k\ + 2k2. + pkp - N and

F (ko k \ kp

-£,0,0,.)>a.

It is proved that for this it suffices to require that the function φ be non-extremal, compact and continuous in the metric p(x, y), and also that for at least one point x satisfying (0.9) for some e > 0 there exists a finite moment degree 1 + e and xn > 0 for any v = 0.1,.

In the second chapter, we study the rough (up to logarithmic equivalence) asymptotics of the probability of large deviations of functions from D = (^0) ■ ) T"n, 0, .) - the number of cells with a given filling in the central region of the parameters N, n. Rough The asymptotics of the probabilities of large deviations is sufficient for studying the indices of the goodness of fit tests.

Let the random variables ^ in (0.2) be identically distributed and

P(z) - generating function of a random variable - converges in a circle of radius 1< R < оо. Следуя /38/, для 0 < z < R обозначим через £(z) случайную величину такую, что

Ml+£ = £ i1+ex„< 00.

0.10) k] = Pk, k = 0.1,.

Denote

If there is a solution of the equation m Z(z) = ъ, then it is unique /38/. Everywhere below we will assume that pk > 0,A; = 0.1,.

In the first paragraph of the first paragraph of the second chapter, there is an asymptotics of the logarithms of probabilities of the form

npP(/x0 = ko,., cp = kn).

The following theorem is proved.

Theorem 2. A rough local theorem on the probabilities of large deviations. Let n, N -» oo so that jj -> 7,0<7 < оо, существует z7 - корень уравнения M£(z) = 7, с. в. £(г7) имеет положительную дисперсию. Тогда для любого k G Cl(n,N)

lnP(A = k) = JftpK)) + O(^lniV).

The statement of the theorem follows directly from the formula for the joint distribution fii,. fin in /26/ and the following estimate: if non-negative integer values ​​, Нп satisfy the condition

Hi + 2d2 + + PNp = n, then the number of nonzero values ​​among them is 0(l/n). This is a rough estimate that does not claim to be new. The number of non-zero zg in generalized layouts does not exceed the value of the maximum filling of cells, which in the central region with a probability tending to 1 does not exceed the value O(lnn) /25/,/27/. Nevertheless, the resulting estimate 0(y/n) is satisfied with probability 1, and it is sufficient to obtain a rough asymptotics.

In the second paragraph of the first paragraph of the second chapter, the value of the limit is found where adz is a sequence of real numbers converging to some a G R, φ(x) is a real-valued function. The following theorem is proved.

Theorem 3. A rough integral theorem on the probabilities of large deviations. Let the conditions of Theorem 2 be satisfied, for some r > 0, C > 0 the real function φ(x) is compact, uniformly continuous in the metric p on the set

A = 0r+<;(p(z7)) П Ц7+с] и удовлетворяет условию неэкстремальности на множестве fly. Если для некоторой константы а такой, что inf ф(х) < а < sup ф(х). xeily существует вектор ра € fi7 П 0r(p(z7)); такой, что

Ф(ra) > a and j(( (x) >a,xe n7),p(2;7)) = 7(pa,p(*y)) mo for any sequence a^ converging to a,