Tree cult of the Turks has been practised for centuries, but cult under fruit trees like pear-tree has not been noticed. In this article traces of the pear-tree cult in the Caucasus is outlined. The Karachay-Balkar Turks living in the Caucasus denote pear with the word kertme, while it is not so in the majority of Turkic languages. This word was borrowed by Hungarians most probably around the Kuban river north to the Caucasus before the conquest of the Carpathian Basin, their later homeland. Hungarians might also have acquired the worship of the pear-tree that time and in that area, because already in the first written sources (in the form of family- and place names) it is well documented. Hints of the pear cult can also be seen in the children's songs in connection with the 'pear-tree'. We can state that the pear-tree cult was known in the Kuban region before 680-700 A.D., for the Hungarians left for Etelköz in those years and did not return there ever after.
Sample path properties of the Cauchy principal values of Brownian and random walk local times are studied. We establish LIL type results (without exact constants). Large and small increments are discussed. A strong approximation result between the above two processes is also proved.