In 1986, Boguslaw Tomaszewski asked the following question: Consider n real numbers a1, . . . , an such that the sum of their squares is 1. Of the 2n expressions |ε1a1 + · · · + εnan| with εi = ±1, can there be more with value > 1 than with value ≤ 1? Apart from being of intrinsic interest in probability, an answer to this conjecture would also have applications in quadratic programming. However, even after more than thirty years the conjecture is still unsolved. In this thesis we settle a special case of the conjecture - we prove that the conjecture holds for vectors of the form (α, δ, . . . , δ) of sufficiently large dimension. This generalizes earlier result which showed that the conjecture holds for vectors of the form (δ, . . . , δ). 1

In this thesis we present some related problems about the renewal processes. More precisely, let $gamma_{t}$ be the residual life at time $t$ of the renewal process $A={A(t),t geq 0}$, $F$ be the common distribution function of the inter-arrival times. Under suitable conditions, we prove that if $Var(gamma_{t})=E^2(gamma_{t})-E(gamma_{t}),forall t=0,1 ho,2 ho,3 ho,... $, then $F$ will be geometrically distributed under the assumption $F$ is discrete. We also discuss the tails of random sums for the renewal process. We prove that the $k$ power of random sum is always new worse than used ($NWU$).