## 1 Introduction  
From both viewpoints of economical design of mechanical structures and reduction of the environmental load, elongation of the design life for every product in the wide range of industries, is an important subject. In such a circumstance, for train axles and wheels, load bearing parts of automobiles, turbine rotors of the power plants and so on, the number of loading cycles reaches over \(10^{9}\) cycles, during the service period of the respective mechanical components [1, 2, 3, 4]. One of the typical aspects of fatigue property in the very high cycle regime for structural steels is the fact that the duplex \(S\)-\(N\) characteristics consisting of the respective \(S\)-\(N\) curves for the surface-initiated and the interior-initiated failures are observed, as reported by many researchers [3, 5, 6, 7]. The term \"\(S\)-\(N\) curve\" refers to the relationship between the stress level [5] and the number of stress cycles to failure [\(N\)]. Expressions of \(S\)-\(N\) property and \(S\)-\(N\) characteristics are also used in this paper, based on a similar sense. In the case of interior-initiated failure, an inclusion is usually found at the core portion of the fish-eye formed on the fracture surface [3, 5, 8]. Thus, the interior inclusion at the crack initiation site plays a dominant role to govern the fatigue strength and fatigue life of the specimen.  
It is well-known that the fatigue limit of the metallic material can be evaluated by combining the material hardness and the concept of \(\sqrt{area}\) for the defect size proposed by [PERSON] et al. [9]. Another important factor to control the fatigue property is the location of the inclusion, since the stress distribution across the section has a steep slope in rotating bending [10]. Accordingly, the fatigue property of those metallic materials should be analyzed as a function of the size and the location of the interior inclusions, together with the hardness. However, both the size and the location of the inclusions inside the material have particular distributions, depending on the fabrication process of the actual metallic materials [11, 12, 13].  
From this point of view, the authors attempted to construct a probabilistic model on the very high cycle fatigue property for the bearing steel of SUJ2 in rotating bending, by combining the distribution characteristics of the size and the location of the interior inclusion. In this model, a two-parameter Weibull distribution was accepted for the distribution pattern of the inclusion size, whereas the random distribution was accepted for the location of the inclusion. Since the scatter of the hardness is sufficiently small, the hardness of this steel was supposed to be constant for the sake of simplicity. Although some number of papers analyzing the statistical aspects of _S-N_ property were published [14; 15; 16; 17; 18; 19], the main target of those research is the statistical property within the usual life region, which is less than \(10^{7}\) cycles. Considering this point, the main aim of the present study was set to the physical interpretation of the statistical aspects of the _S-N_ characteristics of a bearing steel, in the very high cycle regime.