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×ØÖ Øº The range of random walks means the number of distinct sites visited at least once by the random walk before time n. In two or more dimensional cases, we established the law of large numbers for the range of simple symmetric random walks under the conditional probability given the event that the last point is the origin. Moreover we studied the large deviations in the upward direction and obtained the similar results to the original random walk. 2000 Mathematics Subject Classification.doi:10.2748/tmj/1163775134 fatcat:i67bgcgyzrbyxnfvzjzbad5oz4