Asymptotic notation for program execution time. The lower and upper bounds are asymptotically sharp. Sum rule and 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 criteria for testing the possibly complex null hypothesis, which is that for a random sequence such that

Xi е hi, i \u003d 1,, n, where hi \u003d (0,1,., M), for any i \u003d 1,., N, and for any k £ 1m the probability of the event

Xi \u003d k) does not depend on r. This means that the sequence is stationary in a sense.

In a number of applied problems, as a sequence (Xr-) ™ \u003d 1, we consider a sequence of colors of balls when choosing without returning until exhaustion from an urn containing ni - 1\u003e 0 balls of color k, k € 1m. 1,., Pm - 1). Let the urn contain n - 1 balls, m k \u003d 0

We denote by r (k) (fc) Jk) rw - Г! ,. ... ... , 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 the places of adjacent balls of color k in such a way that

Pk Kf \u003d p. 1\u003e \u003d 1

The set of sequences h (fc) for all k £ 1m uniquely determines the sequence. The sequences hk for different k are dependent on each other. 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 neighboring 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 vectors h (n, N ) \u003d (hi,., hjf) with positive integer components such that K \u003d A. (0.1)

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

Having given a certain probability distribution on the set of vectors Н Hn, dg, 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 nonnegative integer components satisfying (0.1). As probability distributions on the set of vectors in the thesis, distributions of the form

P (%, N) \u003d (n,., RN)) \u003d P (£ „\u003d ru, v \u003d l,., N \\ jr ^ \u003d n), (0.2) where. , £ dr are independent non-negative integer random variables.

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

Ri \u003d. , Л ", V \u003d \\,., N.

L \\ +. ... ... + AN

If the random variables £ b\u003e & 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) \u003d (hi,., / Rdr) in generalized schemes for allocating n particles over N cells is occupied by criteria based on based on statistics like 1 m (N -l, nN) \\ N

LN (h (n, N)) \u003d Zfv (hv)

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

0.4) where fu, v \u003d 1,2 ,. and φ are some real-valued functions, N

Mr \u003d E \u003d r), r \u003d 0.1 ,. 1 / \u003d 1

The values \u200b\u200bin / 27 / were called the number of cells containing exactly g particles.

Statistics of the form (0.3) in / 30 / are called separable (additively separable) statistics. If the functions / „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

Е ДМ \u003d Е ДФг (0.5) it follows that the class of symmetric separable statistics of hv coincides with the class of linear functions of fir. Moreover, the class of functions of the form (0.4) is broader than the class of symmetric separable statistics.

But \u003d (#o (n, N)) is a sequence of simple null hypotheses that the distribution of the vector h (n, N) is (0.2), where the random variables ,. in (0.2) are identically distributed and k) \u003d pk, k \u003d 0,1,2,., the parameters n, N change in the central region.

Consider some P £ (0,1) and a sequence of, generally speaking, complex alternatives

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

РШ\u003e an, N (P))\u003e Р

We will reject the hypothesis Hq (ti, N) if fm\u003e asm ((3). If there is a limit

Rn ~ 1nP (0nr\u003e an, N (P)) \u003d u (p, H), where the probability for each N is calculated under the hypothesis Ht (n, N), then the value ^ (/ 3, H) is named in / 38 / the index of the criterion φ at the point (j3, H). The latter limit may, generally speaking, not exist. Therefore, in the dissertation work, in addition to the criterion index, the value

Ish (~ 1pR (fm\u003e al (/?)))

JV-\u003e oo N-oo mean, respectively, the lower and upper limits of the sequence (odr) as N -\u003e oo,

If a criterion index exists, then the criterion subscript coincides with it. The criterion subscript always exists. The higher the value of the criterion index (the lower criterion index), the better in the considered sense the statistical criterion. In / 38 / the problem of constructing goodness-of-fit criteria for generalized layouts with the greatest value the index of the criterion in the class of criteria that reject the hypothesis Ho (n, N) at / MO Ml Mtch RG iV "iV" "" "" ~ yv "" ^ "where m\u003e 0 is some fixed number, the sequence of constants u 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 criterion 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 / (ξ) is determined by the corresponding information distance of the Kul-Bak - Leibler - Sanov (the random variable rj satisfies the condition< 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 Cramer's condition, remained open. This did not allow us to finally solve the problem of constructing criteria for testing hypotheses in generalized allocation schemes with the highest rate of convergence to zero for the probability of a type I error 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 the specified problem.

The aim of the thesis is to construct goodness-of-fit criteria with the highest criterion index (lower criterion index) for testing hypotheses in a selection scheme without returning in the class of criteria that reject the hypothesis U (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 the entropy and information distance of Kul-bak - Leibler - Sanov 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 Cramer's condition;

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

Scientific novelty:

Scientific and practical value. A number of questions about the behavior of the probabilities of large deviations in generalized allocation schemes have been 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 testing a hypothesis by a single sequence of colors of balls from the fact that this sequence was obtained as a result of a choice without returning until exhaustion of balls from an urn containing balls of two colors, and each such choice has the same probability, to the construction of goodness-of-fit tests for testing hypotheses in the corresponding generalized layout scheme;

Continuity of the Kullback - Leibler - Sanov entropy and information distance functions 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 for symmetric separable statistics that do not satisfy the Cramer condition in the generalized allocation scheme in the semi-axial case;

The 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 test for testing hypotheses in generalized layouts with the highest index value in the class of criteria of the form (0.7).

Approbation of work. The results were presented at the seminars of the Department of Discrete Mathematics of the Mathematical Institute. VA Steklov RAS, Information Security Department of ITMiVT named after S. A. Lebedev RAS and:

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 the Institute of Mathematics and Mechanics and Technology of the Russian Academy of Sciences. SA 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 R&D stage / 21 /. Some of the 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 Physics and Mathematics A.F. Ronzhin, and to the scientific adviser, Doctor of Physics and Mathematics, Senior Researcher A.V. Knyazev. The author expresses his gratitude to Professor A.M. Mathematical Sciences Kruglov IA for the attention shown to this work, and a number of valuable remarks.

The structure and content of the work.

The first chapter examines 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 \u003d (xq, x \\,.) Is an infinite-dimensional vector with a countable number of components;

H (x) - -Ex ^ oXvlnx, -, truncm (x) \u003d (x0, x1,., Xm, 0,0 ,.)] f2 * \u003d (x, xu\u003e 0, zy \u003d 0,1 ,. , oh x „< 1}; Q = {х, х, > 0, u \u003d 0,1,., О xv \u003d 1); \u003d (x G 0, £ L0 \u003d 7);

Ml \u003d o Ue\u003e 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, then for e\u003e 0 by Oe (y) we will denote the set

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

In the second section of the first chapter, a theorem on the boundedness of the entropy of discrete distributions with bounded mathematical expectation is proved.

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

For any w 6 P7

H (x)

If x € fly corresponds to a geometric distribution with a mathematical definition of 7, that is, 7 x „\u003d (1-p) p \\ v \u003d 0.1,., Where p \u003d -,

1 + 7 then the equality

H (x) \u003d F (<7).

The statement of the theorem can be viewed as the result of a formal application of the Lagrange conditional multiplier method 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, has given a rigorous proof.

In the third section 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\u003e 0 with the property yie ~ £<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 a 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. Let J (x, y) denote the information distance

00 £ J (x, y) \u003d Е In-.

Hereinafter, it is assumed that 0 In 0 \u003d 0.0 In jj \u003d 0. The information distance is defined for such x, y such that xn \u003d 0 for all and such that yi \u003d 0. If this condition is not satisfied, then we will put J (x, ij) \u003d oo. Let L SP. Then we will denote

J (A Y) \u003d | nf J (x, y).

In the fourth section of the first chapter, we define 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 a function can be approximated by the values \u200b\u200bof 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

R e then for any 0<7<оо,г>0 the function x) \u003d J (x, p) is compact on the set Ц7] Og (p).

In the fifth section of the first chapter, the properties of the information distance defined on an infinite-dimensional space are considered. Compared with 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) \u003d E \\ Xi ~ Y \\, u \u003d 0

E (xv - Yi) 2 v \u003d Q

P3 (x, y) \u003d 8Up \\ xu-yv \\. v

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

1. For any x, x "€ fi

H (x) - H (x ") \\< - 1){Н{х) + Н{х")).

2. If for some x, p e there exists an e\u003e 0 such that x 6 0 £ (p), then for any x "e Q J (x, p) - J (x", p) |< (е"М - 1){Н{х) + Н{х") + ееН(р)).

Taking into account Theorem 1, these inequalities imply the uniform continuity of the entropy and information distance functions 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

THEN for any 0<7<оои£>0 function

A p (x) \u003d J (x, p) is uniformly continuous on the set Oe (p) in the metric p (x, y).

The definition of non-extremal function is given. The non-extremity condition means that the function has no local extrema, or the function takes the same values \u200b\u200bat 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-extremity condition.

Let for some 7\u003e 0, the region A is given by the condition

A \u003d (x € VLv4\u003e (x)\u003e 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 the parameters n, N in the central region, ^ -; 7, for all their sufficiently large values \u200b\u200bthere are non-negative integers ko, k1,., Kn such that k0 + ki +. + kn \u003d N, k \\ + 2k2. + pkp - N and

Ф (ko k \\ kn

- £, 0,0,.)\u003e A.

It is shown that for this it is sufficient for the function φ to 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 ε\u003e 0 there exists a finite moment of degree 1 + e and x „\u003e 0 for any v \u003d 0,1 ,.

In the second chapter, a rough (up to logarithmic equivalence) asymptotic behavior of the probability of large deviations of functions from A \u003d (^ 0) ■) C "n, 0,.) Is investigated - the number of cells with a given filling in the central region of variation of the parameters N, n. the asymptotics of the probabilities of large deviations are sufficient for studying the indices of the goodness-of-fit tests.

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

P (z), the 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 + £ \u003d £ i1 + ex „< 00.

0.10) k] \u003d Pk, k \u003d 0.1 ,.

We denote

If there is a solution to the equation m Z (z) \u003d b then it is unique / 38 /. Throughout what follows we will assume that pk\u003e 0, A; \u003d 0.1 ,.

In the first section of the first section of the second chapter, we find the asymptotics of the logarithms of the probabilities of the form

1nP (/ x0 \u003d ko,., Cn \u003d 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 -\u003e 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)

1nP (A \u003d k) \u003d JftpK)) + O (^ lniV).

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

Hi + 2d2 + + PNn \u003d n, then the number of nonzero values \u200b\u200bamong them is 0 (n / n). This is a rough estimate that does not claim to be new. The number of nonzero μg in generalized allocation schemes 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 obtained 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 adr 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\u003e 0, C\u003e 0 the real function φ (x) is compact, uniformly continuous in the metric p on the set

A \u003d 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 (pa)\u003e a and j (( (x)\u003e a, xe 7), p (2; 7)) \u003d 7 (pa, p (* y)) mo for any sequence a ^ converging to a,