Advances in Systems Science and Applications (2013) Vol.13 No.1 53-67 Soft Probability of Large Deviations D.A. Molodtsov Dorodnitsyn Computing Center, Russian Academy of Sciences, Moscow, Russia Abstract A concept of soft probability is presented. An analogue of Chebyshev’s inequality for soft probability is proved. Soft large deviation probabilities for a nonnegative random variable under a mean hypothesis is calculated. Keywords soft probability, hypothesis replacement, soft Chebyshev inequality, large deviations. 1 What is Soft Probability? In modern textbooks of probability theory, the exposition usually begins with a discussion of the subject matter of this science. All phenomena are divided into three types. Phenomena of the first type are those characterized by deterministic regularity; this means that a given set of circumstances always leads to the same outcome. Phenomena of the second and the third type are not deterministic regular. The second type consists of statistically regular phenomena, and the third type, of the remaining phenomena. By statistical regularity the statistical stability of outcome frequencies is usu- ally understood. As a rule, this notion is associated with the example of coin tossing. Some authors outline more constructive ways of interpretation [1], but a final formalization of statistical stability has never been proposed, and its verifi- cation is always left to the reader’s judgment and intuition. Thus, the most important question of the applicability of the theory to real phenomena remains essentially unanswered. Soft probability [2-7] is merely the logical completion of the construction of statistical regularity, whose approximate description is usually contained in prob- ability theory textbooks. Our approach is based on examine the conclusions to which the logical devel- opment of the notion of statistical regularity leads. First, we mention at once that the verification of statistical regularity suggest- ed here requires a certain set of trial outcomes, that is, a statistical database. If there are no trial results, then there is no object of examination. 2 Statistical Database First, we introduce the base outcome space Ω. Each trial is associated with an element of the set Ω, namely, the outcome of this trial. A statistical database is merely a finite sequence of outcomes: Base = {ω1, ..., ωn}, ωi ∈ Ω. 54 D.A Molodtsov: Soft Probability of Large Deviations By an event A we mean a subset of Ω : A ⊆ Ω. We say that an event occurs under a certain trial if the outcome of this trial belongs to the given event. We have to define the statistical regularity of the occurrence of an event A. In effect, statistical regularity means the closeness of frequencies for a given set of samples. Thus, in formalizing this concept, we must begin with specifying a set of samples for a statistical database. We identify a sample I with the positions in the sequence Base = {ω1, ..., ωn} occupied by its elements; in other words, a sample is a subset I ⊆ Ind(Base) of the set Ind(Base) = {1, ..., n}. In specifying a set of samples, it is natural to first constrain their size; thus, we introduce a parameter determining the size of a sample. As admissible samples we consider any samples of size m which consists of consecutive elements of the set Ind(Base) and are not too “old”. We denote the set of such samples by S(Base,m, τ): S(Base,m, τ) = {(i, i+ 1, ..., i+m− 1), i = τ, ..., N −m+ 1}. Apparently, the set S(Base,m, τ) is a minimal set of samples which are nat- ural to consider in defining the notion of statistical regularity. Of course, other definitions of admissible samples are also possible, which lead to different defini- tions of statistical regularity; it is important that the set of admissible samples be precisely specified. We define the frequency of occurrence of an event A in a sample I as µ(Base, χ(A, ·), I) = 1 |I| ∑ i∈I χ(A,ωi). Here |I| denotes the cardinality of the set I and χ(A,ω) = {1,ω∈A0,ω /∈A. 3 Statistical Regularity Now, it is natural to understand the statistical regularity of the occurrence of an event A as the closeness of the frequencies of occurrence of A in any admissible samples from the set S(Base,m, τ). Below we give a more formal definition of this notion. Definition 1. An event A is said to be statistically (m, τ, δ) − regular on a database Base if |µ(Base, χ(A, ·), I)− µ(Base, χ(A, ·), J)| ≤ δ. for any samples I, J ∈ S(Base,m, τ). We see that the definition of statistical regularity involves several parameters. Apparently, it is for this reason that this notion has not been used, because the Advances in Systems Science and Applications (2013) Vol.13 No.1 55 dependence of statistical regularity on parameters inevitably makes probability dependent on parameters as well, while probability is traditionally thought of as a single real number in the unit interval. However, the parameterization of probability only means the detailing of the description of the situation under consideration. In different situations, probabilities corresponding to different pa- rameters are important. The description using the same number as probability in all situations is coarser than that using parameterized probability. Parameterized families are used very extensively in theory and practice. Thus, different methods of measuring physical quantities yield different results, which naturally leads to parameterized families. A striking example of such a family is the description of mine deposits in geology. In computational mathematics, it often happens that only approximate solu- tions can be found numerically; such solutions form parameterized families too [2,8]. For dealing with such objects, the notion of a soft set was introduced and the theory of soft sets was developed, which has found numerous applications in var- ious areas of mathematics [2-7, 9-24]. For this reason, the alternative to classical probability considered in this paper is called soft probability. Let us contemplate Definition 1. The statistical database consists of the out- comes of events which have already occurred, while the main purpose of the theory is to produce informative statements concerning future events. Thus, the only interesting aspect of the notion of statistical regularity is its use as a hy- pothesis on the future behavior of trial outcomes. Accordingly, the application of soft probability is divided into two processes: • Verifying hypotheses at each step. • Given a set of accepted hypotheses, constructing estimates, predictions, etc. Note that, in classical probability theory, the former process is virtually absent; the statistical regularity hypothesis is accepted only once, before launching the apparatus of probability theory, after which this hypothesis is neither controlled nor verified anew. Moreover, in classical probability theory, the statistical regu- larity hypothesis is accepted for all possible events together. It remains unclear how to handle situations in which some of the events are statistically regular and the other events are not. The definition of statistical regularity is easy to generalize to a random func- tion. By a random function f we mean any real-valued function defined on the set Ω, that is, any function f : Ω → E, where E is the set of real numbers. An example of such a function is the characteristic function χ(A, ·) of a set. 56 D.A Molodtsov: Soft Probability of Large Deviations The mean value of a random function f on a sample I is defined as µ(Base, f, I) = 1 |I| ∑ i∈I f(ωi). Definition 2. A random function f is said to be statistically (m, τ, δ) − regular on a database Base if |µ(Base, f, I)− µ(Base, f, J)| ≤ δ. for any samples I, J ∈ S(Base,m, τ). Thus, the statistical regularity of an event means simply that the characteristic function of this event is statistically regular. The condition that a random function is statistically regular can be written in a different equivalent form as follows. We set 2a = max I∈S(Base,m,τ) µ(Base, f, I) + min I∈S(Base,m,τ) µ(Base, f, I), and 2b = max I∈S(Base,m,τ) µ(Base, f, I)− min I∈S(Base,m,τ) µ(Base, f, I). It is easy to see that the statistical regularity of f is equivalent to the inequality 2b ≤ δ. For any I ∈ S(Base,m, τ), we have |µ(Base, f, I)−a| ≤ b. This readily implies the equivalence of the statistical regularity of f to the fulfillment of the inequality |µ(Base, f, I) − a| ≤ δ/2, or the inclusion µ(Base, f, I) ∈ [a − δ/2, a + δ/2], for any I ∈ S(Base,m, τ). The inclusion is also equivalent to [ min I∈S(Base,m,τ) µ(Base, f, I), max I∈S(Base,m,τ) µ(Base, f, I)] ⊆ [a− δ/2, a+ δ/2]. This suggests the following natural definition. Definition 3. The interval λ(f,Base,m, τ) = [ min I∈S(Base,m,τ) µ(Base, f, I), max I∈S(Base,m,τ) µ(Base, f, I)] is called the (m, τ)− approximate mean value of the random function f on the database Base. We denote the left and right endpoints of this interval by an underscore and an overscore, respectively: λ(f,Base,m, τ) = [λ(f,Base,m, τ), λ(f,Base,m, τ)]. Advances in Systems Science and Applications (2013) Vol.13 No.1 57 4 Hypotheses on the Behavior of a Random Function On the basis of the notion of the statistical regularity of a random function, we can formulate hypotheses of two types on the future behavior of the values of a random function. In fact, these are hypotheses on the future values of the statis- tical database. Definition 4. A databaseBase is said to be statistically (m, δ)− regular with respect to a random function f if |µ(Base, f, I)− µ(Base, f, J)| ≤ δ. for any samples I, J ∈ S(Base,m, 1). Definition 5. A databaseBase is said to be statistically significantly (m, δ, a)− regular with respect to a random function f if |µ(Base, f, I)− a| ≤ δ. for any samples I ∈ S(Base,m, 1). The difference between Definitions 4 and 5 is in that Definition 4 supposes only the closeness of mean values on any admissible samples, while Definition 5 specifies the number to which these mean values must be close with a given accu- racy. In other words, the mean values must belong to the interval [a− δ.a+ δ], or the approximate mean values of the random function under consideration must belong to [a− δ.a+ δ]. These definitions can also be regarded as hypotheses on certain properties of the approximate means of a random function. Definition 4 specifies only the length of an interval containing the approximate mean, and Definition 5 specifies the boundaries of the entire range of this mean. An analysis shows that these hypotheses are often insufficient for obtaining instructive results. In addition to hypotheses on approximate mathematical ex- pectation, hypotheses describing the deviation of the random function under con- sideration from its mathematical expectation (that is, hypotheses similar to that of the existence of variance) are very useful. Such hypotheses can be formulated in various forms. We give only one version. Together with a random function f , consider the random function equal to the absolute value of the deviation of f from the interval [a− δ.a+ δ], that is, defined by g(ω) = max{|f(ω)− a| − δ, 0}. Definition 6. We say that a database Base satisfies the (m, δ, a,△) − variance hypothesis with respect to a random function f if Base is statisti- cally significantly (m,△, 0) − regular with respect to the random function g, i.e. µ(Base, g, I) ≤ △. 58 D.A Molodtsov: Soft Probability of Large Deviations for any samples I ∈ S(Base,m, 1). The approximate mathematical expectation and approximate variance hy- potheses describe a variable oscillating about a certain interval. It is also natural to consider other hypotheses, which describe the growth, decline, periodicity, and other properties of random variables [6-7]. After classes of hypotheses are chosen, dealing with hypotheses is a dynamical step process. At each step, new realization from the base space (trial outcomes) appear. Thus, at each step, it is required to form a list of accepted hypotheses and perform calculations on the basis of this list. 5 Properties of the Approximate Mean Value of a Random Function Properties of an approximate mean are similar to those of mathematical expec- tation. Inequalities involving intervals are assumed to hold componentwise, that is, at each endpoint of the interval. 1). λ(c,Base,m, τ) = [c, c], c ∈ E. 2). if f(ω) ≥ g(ω) for any ω ∈ Ω, then λ(f,Base,m, τ) ≥ λ(g,Base,m, τ). 3). λ(cf,Base,m, τ) = cλ(f,Base,m, τ), c ∈ E, c ≥ 0. 4). λ(f + c,Base,m, τ) = λ(f,Base,m, τ) + c, c ∈ E. 5). λ(−f,Base,m, τ) = −λ(f,Base,m, τ) = [−λ(f,Base,m, τ),−λ(f,Base,m, τ)]. 6). λ(f + g,Base,m, τ) ⊆ λ(f,Base,m, τ) + λ(g,Base,m, τ). 7). λ(f,Base,m, τ) ⊆ λ(f,Base,m, t), τ ≥ t. 6 An Approximate Variance of a Random Variable In classical probability theory, the variance of a random variable is defined as the mathematical expectation of the squared difference between this variable and its expectation. The mathematical expectation of a random function is merely a number. In the case under consideration, the approximate mean is a parametric fam- ily of intervals which characterizes the random variable on a certain database. Hypotheses involving the notion of statistically significant regularity impose con- straints on approximate means. The intervals describing the constraints are not required to equal the corresponding approximate means. Thus, it is convenient to define approximate variance as the measure of deviation of a random variable from a certain interval rather than from the corresponding approximate mean. Definition 7. The approximate (m, τ, a, δ) − variance of a random function f on a database Base is the (m, τ)− approximate mean of the random function max{|f(ω)− a| − δ, 0}, that is, the interval D(f,Base,m, τ, a, δ) = λ(max{|f(·)− a| − δ, 0}, Base,m, τ). Advances in Systems Science and Applications (2013) Vol.13 No.1 59 The simplest properties of approximate variance are as follows. 1). D(f,Base,m, τ, a, δ) ≥ 0. 2). D(cf,Base,m, τ, ca, cδ) = cD(f,Base,m, τ, a, δ), c ∈ E, c ≥ 0. 3). D(−f,Base,m, τ,−a, δ) = D(f,Base,m, τ, a, δ). 4). D(f+g,Base,m, τ, a+b, δ+γ) ≤ D(f,Base,m, τ, a, δ)+D(g,Base,m, τ, b, γ). 5). D(f+g,Base,m, τ, a+b, δ+γ) ≤ D(f,Base,m, τ, a, δ)+D(g,Base,m, τ, b, γ). 6). D(f,Base,m, τ, a, δ) ≥ D(f,Base,m, τ, a, γ), δ ≤ γ. 7). D(f,Base,m, τ, a, δ) ⊇ D(f,Base,m, γ, a, δ), τ ≤ γ. 7 Chebyshev’s Inequality for Soft Probability Statement 1 (Chebyshev’s inequality). If a random function f is nonnega- tive everywhere on Ω and ε > 0, then λ(χ({ω′ ∈ Ω|f(ω′) ≥ ε}, ·), Base,m, τ) ≤ 1 ε λ(f,Base,m, τ). Proof. If a random function f is nonnegative everywhere on Ω, then, as is easy to see, we have f(ω) ≥ εχ({ω′ ∈ Ω|f(ω′) ≥ ε}, ω) for any ε > 0 and any ω ∈ Ω. Property 2 of approximate mean implies λ(f,Base,m, τ) ≥ ελ(χ({ω′ ∈ Ω|f(ω′) ≥ ε}, ω), Base,m, τ). The approximate means of the characteristic function of a set equals the soft probability of this set; therefore, the soft probability of the set {ω′ ∈ Ω|f(ω′) ≥ ε} is estimated as λ(χ({ω′ ∈ Ω|f(ω′) ≥ ε}, ·), Base,m, τ) ≤ 1 ε λ(f,Base,m, τ) (recall that the inequality is interval). This completes the proof of the statement. Now, let f be an arbitrary random function. For the function |f |, we have λ(χ({ω′ ∈ Ω| |f(ω′)| ≥ ε}, ·), Base,m, τ) ≤ 1 ε λ(|f |, Base,m, τ). Since the inequality |f(ω′)| ≥ ε is equivalent to |f s(ω′)| ≥ εs, where s > 0, it follows that λ(χ({ω′ ∈ Ω| |f(ω′)| ≥ ε}, ·), Base,m, τ) ≤ 1 εs λ(|f |s, Base,m, τ). therefore, λ(χ({ω′ ∈ Ω| |f(ω′)| ≥ ε}, ·), Base,m, τ) ≤ inf s>0 1 εs λ(|f |s, Base,m, τ). 60 D.A Molodtsov: Soft Probability of Large Deviations Now, consider the nonnegative random function g(ω) = max{|f(ω)−a|−δ, 0}, where f is an arbitrary random function. Applying Chebyshev’s inequality to g, we obtain λ(χ({ω′ ∈ Ω|max{|f(ω′)−a|−δ, 0} ≥ ε}, ·), Base,m, τ) ≤ 1 ε D(f,Base,m, τ, a, δ). Elementary transformations yield λ(χ({ω′ ∈ Ω||f(ω′)− a| ≥ δ + ε}, ·), Base,m, τ) ≤ 1 ε D(f,Base,m, τ, a, δ). This allows us to write the inequality in the equivalent form λ(χ({ω′ ∈ Ω||f(ω′)− a| ≥ δ}, ·), Base,m, τ) ≤ inf 0<ε<δ 1 ε D(f,Base,m, τ, a, δ − ε). Thus, Chebyshev’s inequality gives an estimate of the soft probability of “large” deviations in terms of approximate mean or approximate variance. If the database is known, then this information is of little value, because it is easy to directly calculate the exact values of any probabilistic characteristics, including the prob- abilities of large deviations. Apparently, it is of more interest to apply this in- equality to estimating the probability of a random function on a future database. Naturally, this requires hypotheses on the approximate mean or the approxi- mate variance of the function under consideration. However, in the presence of hypotheses, of interest are sharp bounds for the probability of large deviations. 8 Soft Probability of Large Deviations for a Nonnegative Random Function Under an Approximate Mean Hypothesis Suppose that a database Base is statistically significantly (m, δ, a) − regular with respect to a nonnegative random function f , i.e., given any sample I ∈ S(Base,m, 1), we have |µ(Base, f, I)− a| ≤ δ We assume that a ≥ 0. We define a large deviation as the event A(f, ε) = {ω ∈ Ω|f(ω) ≥ ε}, where ε > 0. We are interested in the range of values of the soft probability of this event under the above hypothesis. Clearly, the solution essentially depends on the range of the function f . Consider the case where f(Ω) = E+ = {x ∈ E|x ≥ 0}. Let f(ωi) = xi ∈ E+. Then the constraints on the database can be written as constraints on the vector x = {x1, ...xn} ∈ X(n,m, a, δ), where X(n,m, a, δ) = {x ∈ En +| | j+m−1∑ i=j xi − am| ≤ δm, j = 1, ..., n−m+ 1}. Advances in Systems Science and Applications (2013) Vol.13 No.1 61 The boundaries of the range of the soft probability are u∗(n,m, a, k, τ, δ, ε) = sup x∈X(n,m,a,δ) max τ≤j≤n−k+1 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. and u∗(n,m, a, k, τ, δ, ε) = inf x∈X(n,m,a,δ) min τ≤j≤n−k+1 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. Note that, at n = m, the set X(n,m, a, δ) takes the form X(m,m, a, δ) = {x ∈ Em + | | m∑ i=1 xi − am| ≤ δm}. In the case k ≤ m, the evaluation of u∗ and u∗ is based on the following assertion. Statement 2. If x ∈ Em + and (xj , xj+1, ..., xj+m−1) ∈ X(m,m, a, δ) , then there exists a vector y ∈ X(n,m, a, δ) such that yi = xi for i = j, j+1, ..., j+m−1. Proof. Let yi = xj+(i−j)modm for i = 1, ..., n. Then ∑l+m−1 i=l yi = ∑j+m−1 i=j xi for any l = 1, ..., n−m+1. Therefore, y ∈ X(n,m, a, δ). This proves the required assertion. The u∗ and u∗ problems can be formulated as u∗(n,m, a, k, τ, δ, ε) = max τ≤j≤n−k+1 sup x∈X(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. and u∗(n,m, a, k, τ, δ, ε) = min τ≤j≤n−k+1 inf x∈X(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi)}. At k ≤ m, in the problems sup x∈X(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi) and inf x∈X(n,m,a,δ) 1 k j+k−1∑ i=j χ({y|y ≥ ε}, xi) the function to be optimized depends only on the variables xj = (xj , xj+1, xj+m−1); hence, we can perform optimization over the projection of the set X(n,m, a, δ) 62 D.A Molodtsov: Soft Probability of Large Deviations on the corresponding coordinates rather over the entire set. It follows from State- ment 2 that this projection coincides with the set X(m,m, a, δ); thus, at k ≤ m, the u∗ and u∗ problems take the forms u∗(n,m, a, k, τ, δ, ε) = sup x∈X(m,m,a,δ) 1 k k∑ i=1 χ({y|y ≥ ε}, xi)}. and u∗(n,m, a, k, τ, δ, ε) = inf x∈X(m,m,a,δ) 1 k k∑ i=1 χ({y|y ≥ ε}, xi)}. For x ∈ Em + , we set π(x,m, ε) = |{i ∈ {1, ...,m}|xi ≥ ε}|; this is the number of components greater than or equal to ε. We have u∗(n,m, a, k, τ, δ, ε) = sup x∈X(m,m,a,δ) min{π(x,m, ε), k} k = 1 k min{ sup x∈X(m,m,a,δ) π(x,m, ε), k}. and u∗(n,m, a, k, τ, δ, ε) = 1 k max{k + inf x∈X(m,m,a,δ) π(x,m, ε)−m, 0}. Thus, it is required to find the maximum and the minimum value of the function π(x,m, ε) on X(m,m, a, δ). Let us introduce the set Π(m, ε, p) = {x ∈ Em + |π(x,m, ε) = p}. It is easy to see that the image of the function ∑m i=1 xi on the set Π(m, ε, p) equals { [εp,+∞), p > 0 [0,mε), p = 0, Therefore, the function π(x,m, ε) takes the value p on the set X(m,m, a, δ) if and only if Π(m, ε, p) ∩X(m,m, a, δ) ̸= ∅. that is, 1 ≤ p ≤ ma+mδ ε or a− δ < ε at p = 0. Let [x] denote the largest integer not exceeding x. Then the condition on those positive values p which the function π(x,m, ε) can take on X(m,m, a, δ) can be written in the form 1 ≤ p ≤ min{m, [ m(a+ δ) ε ]}. Advances in Systems Science and Applications (2013) Vol.13 No.1 63 The condition that π(x,m, ε) vanishes on X(m,m, a, δ) has the form a < δ + ε. Thus, we have proved the following assertion. Statement 3. Let k ≤ m. 1). If 0 ≤ a < δ + ε, then u∗(n,m, a, k, τ, δ, ε) = 0. 2). If a > δ + ε, then u∗(n,m, a, k, τ, δ, ε) = max{k+1−m,0} k = {1/m,k=m 0,k m. The function ∑j+k−1 i=j χ({y|y ≥ ε}, xi) depends only on those components of the vector x whose numbers belong to {j, j+1, ..., j+ k − 1}. Thus, we introduce the set X(k,m, a, δ) = {x ∈ Ek +| | j+m−1∑ i=j xi − am| ≤ δm, j = 1, ..., k −m+ 1}. For this set, an assertion similar to Statement 2 is valid. Statement 4. If x ∈ En + and (xj , xj+1, ..., xj+k−1) ∈ X(k,m, a, δ), then there exists a vector y ∈ X(n,m, a, δ) such that yi = xi for i = j, j + 1, ..., j + k − 1. Proof. We set • yi = xj+(i−j)mod m for i = 1, ..., j − 1, • yi = xi for i = j, j + 1, ..., j + k − 1, • yi = xj+k−m+(i−j−k+m)mod m for i = j + k, ..., n. We have • ∑l+m−1 i=l yi = ∑j+m−1 i=j xi for l = 1, ..., j − 1, • ∑l+m−1 i=l yi = ∑l+m−1 i=l xi for l = j, j + 1, ..., j + k −m, • ∑l+m−1 i=l yi = ∑j+k−1 i=j+k−m xi for l = j + k −m, ..., n−m+ 1. Therefore, y ∈ X(n,m, a, δ). This completes the proof of the statement. Now, the problems for u∗ and u∗ with k > m take the forms u∗(n,m, a, k, τ, δ, ε) = 1 k sup x∈X(k,m,a,δ) π(x, k, ε). and u∗(n,m, a, k, τ, δ, ε) = 1 k inf x∈X(k,m,a,δ) π(x, k, ε). Let us divide k by m with a remainder, that is, write k = mq + r,m > r ≥ 0. Take an arbitrary vector x ∈ X(k,m, a, δ) and consider its decomposition into the parts x0 = (x1, ..., xm) ∈ Em and xj = (xr+(j−1)m+1, xr+(j−1)m+m) ∈ Em, j = 1, ..., q. 64 D.A Molodtsov: Soft Probability of Large Deviations It is assumed that r > 0; if r = 0, then the vector x0 is absent. Obviously, π(x, k, ε) = π(x0, r, ε) + q∑ j=1 π(xj ,m, ε) ≤ min{π(x0,m, ε), r}+ q∑ j=1 π(xj ,m, ε). Note that the condition x ∈ X(k,m, a, δ) implies that xj ∈ X(m,m, a, δ) for any j = 0, ..., q. Hence, we have sup x∈X(k,m,a,δ) π(x, k, ε) ≤ min{ max y∈X(m,m,a,δ) π(y,m, ε), r}+ q max y∈X(m,m,a,δ) π(y,m, ε). Let us show that this inequality is, in fact, an equality. We take a vector y∗ ∈ X(m,m, a, δ) at which max y∈X(m,m,a,δ) π(y,m, ε) is attained and place all components of this vector greater than or equal to ε at the first positions. Let x∗ ∈ Ek + be the vector formed by as many copies of y∗ written one after another as needed to achieve the required dimension. It is easy to see that x∗ ∈ X(k,m, a, δ) and sup x∈X(k,m,a,δ) π(x, k, ε) ≥ π(x∗, k, ε) = min{ max y∈X(m,m,a,δ) π(y,m, ε), r} + q max y∈X(m,m,a,δ) π(y,m, ε). Thus, we have proved the following assertion. Statement 5. Suppose that n ≥ k > m and k = mq + r, m > r ≥ 0 . 1). Ifm(a+δ) ≥ ε, then u∗(n,m, a, k, τ, δ, ε) = min{[m(a+δ) ε ],r}+q min{[m(a+δ) ε ],m} k . 2). If m(a+ δ) ≤ ε, then u∗(n,m, a, k, τ, δ, ε) = 0. It is easy to see that Statement 5 is also valid for n ≥ m ≥ k > 0. Now, consider the u∗ problem. For the arbitrary vector x ∈ X(k,m, a, δ) under consideration and its partition constructed above, we have π(x, k, ε) = π(x0, r, ε)+ q∑ j=1 π(xj ,m, ε) ≥ max{k−m+π(x0,m, ε), 0}+ q∑ j=1 π(xj ,m, ε). Since xj ∈ X(m,m, a, δ) for any j = 0, ..., q, it follows that inf x∈X(k,m,a,δ) π(x, k, ε) ≥ max{r−m+ inf y∈X(m,m,a,δ) π(y,m, ε), 0}+q inf y∈X(m,m,a,δ) π(y,m, ε). As above, this inequality is, in fact, an equality. To show this, we take a vector y∗ ∈ X(m,m, a, δ) at which min y∈X(m,m,a,δ) π(y,m, ε) is attained and place all com- ponents of this vector which are greater than or equal to ε at the last positions. Advances in Systems Science and Applications (2013) Vol.13 No.1 65 Consider the vector x∗ ∈ Em + consisting of copies of y∗ written one after another. It is easy to see that x∗ ∈ X(k,m, a, δ) and inf x∈X(k,m,a,δ) π(x, k, ε) ≤ π(x∗, k, ε) = max{r −m+ min y∈X(m,m,a,δ) π(y,m, ε), 0}+ q min y∈X(m,m,a,δ) π(y,m, ε). Thus, we have proved the following assertion. Statement 6. Suppose that n ≥ k > m and k = mq + r,m > r ≥ 0 . 1). If 0 ≤ a < δ + ε, then u∗(n,m, a, k, τ, δ, ε) = 0. 2). If a ≥ δ + ε, then u∗(n,m, a, k, τ, δ, ε) = q k . 9 Conclusion The exact boundaries of the range of the soft probability of large deviations under a single mean hypothesis, which were found in this paper, show (although, for a very simple example) that it is quite possible to deal with soft probabilities, in spite of the presence of parameters and the interval form of soft probability. The next goal is to solve more complicated problems on evaluating various probabili- ties and other characteristics in the presence of several hypotheses, preferably of different types. Of special interest is the application of the ideas and results presented in this paper to a real statistical problem, which would make it possible to verify the effectiveness of the approach for real data. All readers interested in such a prac- tical experiment are kindly requested to send their suggestions to the author at dmitri molodtsov@mail.ru. References [1] V. N. Tutubalin. (1972), Probability Theory: A Short Course and Scientific- Methodological Notes, Izd. Moskov. Univ, Moscow, Russian. [2] D. A. Molodtsov. (2004), Theory of Soft Sets, URSS, Moscow, Russian. [3] D. A. Molodtsov. (2007), “Portfolio control using soft probability”, Vestn. Nats. Assots. Uchastnikov Fond. Rynka, No.7-8. [4] D. A. Molodtsov. (2007), “Portfolio control using soft probability (short positions)”, Vestn. Nats. Assots. Uchastnikov Fond. Rynka, No.10. [5] D. A. Molodtsov. (2011), “Soft sets and prediction”, Nechetkie Sist. Myagkie Vychisl, Vol.6, No.1. [6] D. A. Molodtsov. (2010), “Finite frequency stability and probability”, Nechetkie Sist. Myagkie Vychisl, Vol.5, No.1. 66 D.A Molodtsov: Soft Probability of Large Deviations [7] D. A. Molodtsov. (2008), “Soft uncertainty and probability”, Nechetkie Sist. Myagkie Vychisl, Vol.3, No.1. [8] I. S. Berezin and N. P. Zhidkov. (1962), Computing Methods, Pergamon Press, Vol.1/2. [9] H. Aktas and N. Cagman N. (2007), “Soft sets and soft groups”, Inf. Sci, Vol.177, pp.2726C2735. [10] M. I. Ali et al. (2009), “On some new operations in soft set theory”, Com- puters and Math. With Appl, Vol.57, pp.1547-1553. [11] D. Chen. ((2005)), “The parameterization reduction of soft sets and its ap- plications”, Comput. Math. Appl, Vol.49, pp.757-763. [12] F. Feng et al. (2008), “Soft semirings”, Computers and Math. With Appl, Vol.56, pp.2621-2628. [13] F. Feng et al. (2010), “Soft sets combined with fuzzy sets and rough sets: a tentative approach”, Soft Computing, Vol.14, pp.899-911. [14] Y. B. Jun. (2008), “Soft BCK/BCI-algebras”, Computers and Math. With Appl, Vol.56, pp.1408-1413. [15] Y. B. Jun and C. H. Park. (2008), “Applications of soft sets in ideal theory of BCK/BCI-algebras”, Inf. Sci, Vol.178, pp.2466-2475. [16] Z. Kong et al. (2008), “The normal parameter reduction of soft sets and its algorithm”, J.Comp. Appl. Math, Vol.56, pp.3029-3037. [17] P. K. Maji et al. (2002), “An application of soft sets in a decision making problem”, Comput. Math. Appl, Vol.44, pp.1077-1083. [18] P. K. Maji et al. (2003), “Soft set theory”, Comput. Math. Appl, Vol.45, pp.555-562. [19] P. Majumdar and S. K. Samanta. (2010), “On soft mappings”, Computers and Math. With Appl, Vol.60, pp.2666-2672. [20] D. Molodtsov. (1999), “Soft set theory first results”, Comput. Math. Appl, Vol.37, pp.19-31. [21] D. Pie and D. Miao. (2005), “From soft sets to information systems, granular computing”, IEEE Iinter. Conf, No.2, pp.617-621. Advances in Systems Science and Applications (2013) Vol.13 No.1 67 [22] M. Shabir et al. (2009), “Soft ideals and generalized fuzzy ideals in semi- groups”, New Math. Nat. Comput, No.5, pp.599-615. [23] M. Shabir et al. (2011), “On soft topological spaces”, Computers and Math. With Appl, Vol.61, pp.1786-1799. [24] Zou Yan et al. (2008), “Data analysis approaches of soft sets under incom- plete information”, Knowl.-Based Syst, Vol.21, pp.941-945. Corresponding author D.A Molodtsov can be contacted at: dmitri molodtsov@mail.ru