On some combinatorial relations concerning the symmetric random walk

Kanwar, Sen (1964) On some combinatorial relations concerning the symmetric random walk. A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (3). pp. 335-357.

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Abstract

In one-dimensional symmetric random walk it is interesting to note how often the particle intersects a given line and also how often it remains above it. I. Vincze and E. Csáki [3] have determined in connection with a statistical problem regarding the Galton-test the distribution of the number of intersections and also the joint distribution of the number of intersections and of the positive steps in the case the particle returns at the end to the origin. In another paper E. Csáki [4] has given the distribution of the number of intersections without assuming to which point the particle returns at the end. In the following paper, I should like to determine the above distributions when the particle reaches at the end some fixed point other than the origin. This problem may be interpreted as: Two players A and В play a coin tossing game in which player A wins or loses a unit amount according to whether the result of the coin tossing is "head" or "tail". Assuming that at the end of the game A leads over В by certain fixed units, we are interested in investigating how often one overtakes the other and also how often A has been leading over B. In this paper, we shall consider the sequences δ ≡ (δ₁, δ₂, . . ., δ₂ₙ) of n + k (+ l)'s and n — k (— l)'s, each possible array has the same probability (2n n - k)⁻¹ = (2n n + k)⁻¹. The partial sum of δᵢ's is denoted by sᵢ, i.e., sᵢ = δ₁ + δ₁ + . . . + δᵢ, (i = 1, 2, . . ., 2n) , s₀ = 0 and s₂ₙ = 2 k . We shall call the array {s₀, s₁, . . ., s₂ₙ} the path of the particle. Thus each array (δ₁, δ₂, . . ., δ₂ᵢ, . . ., δ₂ₙ) corresponds to a random path of the particle starting at the origin and reaching after 2 n steps the point (2 n, 2 k) (0 < k ≦ n). Each path has the same probability. If the points (i, sᵢ) are represented in a plane and each of them is connected with the next one, then we obtain a figure illustrating the path of the particle. In the following, we shall consider the distributions of λ (number of intersections) and γ (number of positive steps).

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